Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624170 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10)(11,22)(12,28)(13,19)(14,20)(15,34)(16,37)(17,25)(18,31)(21,36)(23,50)(24,48)(26,56)(27,40)(29,61)(30,62)(32,45)(33,64)(35,42)(38,46)(39,67)(41,54)(43,63)(44,65)(47,53)(49,58)(51,70)(52,60)(55,66)(57,69)(59,71)(68,72);(1,2)(3,6)(4,5)(7,8)(9,11,14,25)(10,17,20,22)(12,23,30,52)(13,24,16,27)(15,26,33,55)(18,39,43,47)(19,40,37,48)(21,41,45,49)(28,60,62,50)(29,51,38,59)(31,53,63,67)(32,54,36,58)(34,66,64,56)(35,57,65,72)(42,68,44,69)(46,70,61,71);(9,12)(10,18)(11,23)(13,29)(14,30)(15,35)(16,38)(17,39)(19,42)(20,43)(21,46)(22,47)(24,51)(25,52)(26,57)(27,59)(28,34)(31,36)(32,63)(33,65)(37,44)(40,68)(41,70)(45,61)(48,69)(49,71)(50,56)(53,58)(54,67)(55,72)(60,66)(62,64);(2,5)(6,8)(9,13,14,16)(10,19,20,37)(11,24,25,27)(12,29,30,38)(15,32,33,36)(17,40,22,48)(18,42,43,44)(21,34,45,64)(23,51,52,59)(26,54,55,58)(28,61,62,46)(31,35,63,65)(39,68,47,69)(41,66,49,56)(50,70,60,71)(53,57,67,72);(1,3,4,7)(2,6,5,8)(10,20)(11,25)(18,43)(19,37)(21,45)(23,52)(24,27)(26,55)(28,62)(34,64)(42,44)(46,61)(51,59)(53,67)(54,58)(57,72)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531824);