Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| 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 | #23891 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,26)(13,19)(14,35)(15,20)(16,21)(18,30)(22,51)(23,39)(24,41)(25,42)(27,38)(28,57)(29,50)(31,45)(32,46)(33,48)(34,49)(36,40)(37,47)(43,53)(44,60)(52,69)(54,65)(55,64)(56,63)(58,62)(59,68)(61,67)(66,70)(71,72);(2,5)(6,8)(9,11,16,25)(10,17,21,42)(12,27,29,56)(13,32,34,54)(14,24,15,36)(18,43,45,51)(19,46,49,65)(20,40,35,41)(22,30,53,31)(23,37,55,33)(26,38,50,63)(28,52,59,70)(39,47,64,48)(44,62,67,72)(57,69,68,66)(58,61,71,60);(9,12,16,29)(10,18,21,45)(11,22,25,53)(13,28,34,59)(14,31,15,30)(17,38,42,63)(19,44,49,67)(20,50,35,26)(23,52,55,70)(24,56,36,27)(32,58,54,71)(33,61,37,60)(39,62,64,72)(40,43,41,51)(46,66,65,69)(47,68,48,57);(9,13,16,34)(10,19,21,49)(11,23,25,55)(12,28,29,59)(14,33,15,37)(17,39,42,64)(18,44,45,67)(20,47,35,48)(22,52,53,70)(24,54,36,32)(26,57,50,68)(27,58,56,71)(30,60,31,61)(38,62,63,72)(40,46,41,65)(43,66,51,69);(1,3,4,7)(2,6,5,8)(12,29)(18,45)(22,53)(26,50)(27,56)(28,59)(30,31)(38,63)(43,51)(44,67)(52,70)(57,68)(58,71)(60,61)(62,72)(66,69)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346978);