Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32867 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | #578 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,38,6,13,8,15)(3,16,36,44,19,10,21,56)(4,11,37,57,24,47,26,49)(5,27,31,51,29,83,39,12)(7,14,40,95,32,52,35,55)(17,58,72,99,63,41,64,103)(18,65,69,42,67,101,91,59)(20,60,74,53,70,43,33,93)(22,73,80,102,76,45,71,107)(23,77,79,106,28,81,86,46)(25,48,92,111,61,97,34,54)(30,82,90,110,87,127,96,50)(62,114,115,98,66,116,118,112)(68,117,88,100,119,108,123,113)(75,124,121,104,84,125,120,122)(78,109,94,128,85,126,89,105);(1,3,6,19)(2,10,13,16)(4,22,24,76)(5,18,29,67)(7,33,32,74)(8,36,9,21)(11,45,47,73)(12,42,51,59)(14,53,52,93)(15,56,38,44)(17,61,63,25)(20,40,70,35)(23,75,28,84)(26,80,37,71)(27,65,83,101)(30,88,87,123)(31,91,39,69)(34,72,92,64)(41,97,58,48)(43,95,60,55)(46,104,106,122)(49,107,57,102)(50,108,110,117)(54,103,111,99)(62,85,66,78)(68,96,119,90)(77,124,81,125)(79,120,86,121)(82,100,127,113)(89,115,94,118)(98,128,112,105)(109,114,126,116);(1,4,6,24)(2,11,13,47)(3,17,19,63)(5,28,29,23)(7,34,32,92)(8,37,9,26)(10,41,16,58)(12,46,51,106)(14,54,52,111)(15,57,38,49)(18,66,67,62)(20,71,70,80)(21,72,36,64)(22,74,76,33)(25,40,61,35)(27,81,83,77)(30,89,87,94)(31,79,39,86)(42,98,59,112)(43,102,60,107)(44,103,56,99)(45,93,73,53)(48,95,97,55)(50,109,110,126)(65,116,101,114)(68,120,119,121)(69,115,91,118)(75,88,84,123)(78,90,85,96)(82,105,127,128)(100,125,113,124)(104,117,122,108);(1,5,9,39,6,29,8,31)(2,12,15,27,13,51,38,83)(3,18,36,91,19,67,21,69)(4,23,37,86,24,28,26,79)(7,30,40,96,32,87,35,90)(10,42,44,101,16,59,56,65)(11,46,49,77,47,106,57,81)(14,50,55,82,52,110,95,127)(17,62,72,118,63,66,64,115)(20,68,74,123,70,119,33,88)(22,75,80,120,76,84,71,121)(25,78,92,89,61,85,34,94)(41,98,99,116,58,112,103,114)(43,100,53,108,60,113,93,117)(45,104,102,125,73,122,107,124)(48,105,54,109,97,128,111,126)Polycyclic presentation:
2^9|a^2=efh,b^2=i,c^2=i,d^2=fh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418936);