Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509829 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31032 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24296 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,6,13,35,54,88,77,8,12,37,53,7,14,36,58)(3,15,18,55,19,44,74,108,21,9,32,42,20,43,75,107)(4,22,26,59,38,99,97,48,25,79,87,47,27,56,33,10)(5,28,29,50,39,100,98,52,31,57,94,11,30,51,34,60)(16,62,65,103,23,76,81,40,64,102,96,45,66,110,72,61)(17,67,68,41,69,104,95,105,71,91,93,49,70,116,73,106)(24,82,85,78,90,46,111,112,84,114,126,123,86,113,89,92)(63,118,120,117,83,115,125,127,119,109,124,128,80,101,121,122);(1,3)(2,9)(4,23)(5,17)(6,32)(7,20)(8,21)(10,45)(11,49)(12,15)(13,55)(14,44)(16,27)(18,37)(19,35)(22,76)(24,83)(25,66)(26,72)(28,91)(29,93)(30,70)(31,71)(33,96)(34,95)(36,74)(38,64)(39,69)(40,47)(41,50)(42,53)(43,54)(46,109)(48,103)(51,104)(52,105)(56,62)(57,67)(58,107)(59,61)(60,106)(63,86)(65,97)(68,94)(73,98)(75,88)(77,108)(78,122)(79,110)(80,84)(81,87)(82,115)(85,121)(89,124)(90,119)(92,128)(99,102)(100,116)(101,114)(111,120)(112,117)(113,118)(123,127)(125,126);(1,4,8,25)(2,10,12,48)(3,16,21,64)(5,24,31,84)(6,33,37,97)(7,38,35,27)(9,40,15,61)(11,46,50,113)(13,56,53,99)(14,59,54,47)(17,63,71,119)(18,72,32,81)(19,66,20,23)(22,77,79,58)(26,88,87,36)(28,92,57,112)(29,89,94,111)(30,90,39,86)(34,85,98,126)(41,101,49,115)(42,76,55,110)(43,103,44,45)(51,78,100,123)(52,114,60,82)(62,108,102,107)(65,74,96,75)(67,122,91,127)(68,121,93,125)(69,80,70,83)(73,120,95,124)(104,128,116,117)(105,109,106,118);(1,5,8,31)(2,11,12,50)(3,17,21,71)(4,24,25,84)(6,34,37,98)(7,39,35,30)(9,41,15,49)(10,46,48,113)(13,57,53,28)(14,60,54,52)(16,63,64,119)(18,73,32,95)(19,70,20,69)(22,78,79,123)(23,80,66,83)(26,89,87,111)(27,90,38,86)(29,88,94,36)(33,85,97,126)(40,101,61,115)(42,67,55,91)(43,106,44,105)(45,109,103,118)(47,114,59,82)(51,77,100,58)(56,112,99,92)(62,117,102,128)(65,121,96,125)(68,74,93,75)(72,120,81,124)(76,122,110,127)(104,107,116,108)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fh,g^d=gi,h^d=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417028);