Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60462 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10,16,21)(11,17,23,39)(12,26,30,45)(13,19,32,48)(14,34,15,20)(18,36,46,29)(22,43,51,64)(24,40,54,63)(25,42,35,41)(27,56,53,38)(28,57,59,68)(31,47,55,65)(33,50,37,49)(44,61,67,60)(52,66,69,72)(58,71,70,62);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,27)(13,31)(14,25)(15,35)(16,23)(18,43)(19,47)(20,41)(21,39)(22,29)(24,33)(26,56)(28,52)(30,53)(32,55)(34,42)(36,51)(37,54)(38,45)(40,50)(44,62)(46,64)(48,65)(49,63)(57,66)(58,61)(59,69)(60,70)(67,71)(68,72);(9,12,15,36,16,30,14,29)(10,18,34,45,21,46,20,26)(11,22,35,27,23,51,25,53)(13,28,37,61,32,59,33,60)(17,38,42,64,39,56,41,43)(19,44,50,68,48,67,49,57)(24,52,31,58,54,69,55,70)(40,62,65,72,63,71,47,66);(2,6)(5,7)(11,23)(13,32)(17,39)(19,48)(22,51)(25,35)(27,53)(28,59)(33,37)(38,56)(41,42)(43,64)(44,67)(49,50)(57,68)(60,61);(3,8)(5,7)(9,13)(10,19)(11,24)(12,28)(14,33)(15,37)(16,32)(17,40)(18,44)(20,49)(21,48)(22,52)(23,54)(25,55)(26,57)(27,58)(29,60)(30,59)(31,35)(34,50)(36,61)(38,62)(39,63)(41,47)(42,65)(43,66)(45,68)(46,67)(51,69)(53,70)(56,71)(64,72)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590473);