Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22×D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×D4×C16 |
| prime radical | #575 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×M64 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,6,13,7,14,31,48,8,12,32,46,34,47,76,94,9,15,33,49,35,50,77,95,5,27,29,71,30,72,75,111)(3,16,19,52,20,53,59,98,18,10,36,38,58,39,79,84,21,40,60,85,61,87,103,119,28,41,73,86,74,88,112,120)(4,11,24,44,25,45,68,93,26,43,69,91,70,92,110,122,22,42,62,89,63,90,104,121,23,64,66,105,67,106,109,125)(17,51,56,96,57,97,102,123,55,37,78,81,101,82,113,116,54,83,99,117,100,118,124,128,65,80,107,114,108,115,126,127);(1,3)(2,10)(4,17)(5,28)(6,19)(7,20)(8,18)(9,21)(11,37)(12,40)(13,38)(14,39)(15,41)(16,27)(22,54)(23,65)(24,56)(25,57)(26,55)(29,73)(30,74)(31,59)(32,36)(33,60)(34,58)(35,61)(42,80)(43,83)(44,81)(45,82)(46,85)(47,87)(48,84)(49,86)(50,88)(51,64)(52,71)(53,72)(62,99)(63,100)(66,107)(67,108)(68,102)(69,78)(70,101)(75,112)(76,79)(77,103)(89,114)(90,115)(91,117)(92,118)(93,116)(94,119)(95,120)(96,105)(97,106)(98,111)(104,124)(109,126)(110,113)(121,127)(122,128)(123,125);(1,4,6,24,7,25,31,68,8,26,32,69,34,70,76,110,9,22,33,62,35,63,77,104,5,23,29,66,30,67,75,109)(2,11,13,44,14,45,48,93,12,43,46,91,47,92,94,122,15,42,49,89,50,90,95,121,27,64,71,105,72,106,111,125)(3,17,19,56,20,57,59,102,18,55,36,78,58,101,79,113,21,54,60,99,61,100,103,124,28,65,73,107,74,108,112,126)(10,37,38,81,39,82,84,116,40,83,85,117,87,118,119,128,41,80,86,114,88,115,120,127,16,51,52,96,53,97,98,123);(4,22)(11,42)(17,54)(23,26)(24,62)(25,63)(37,80)(43,64)(44,89)(45,90)(51,83)(55,65)(56,99)(57,100)(66,69)(67,70)(68,104)(78,107)(81,114)(82,115)(91,105)(92,106)(93,121)(96,117)(97,118)(101,108)(102,124)(109,110)(113,126)(116,127)(122,125)(123,128)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=i,f^2=g,g^2=h,h^2=i,b^a=be,e^a=ei,e^b=ei,d^c=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419946);