Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| 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 | |
1283 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)(9,10,16,22)(11,17,25,40)(12,28,31,60)(13,21,15,19)(14,33,36,46)(18,43,45,30)(20,32,48,35)(23,50,52,70)(24,42,27,39)(26,56,55,66)(29,61,37,47)(34,49,62,44)(38,63,65,53)(41,54,67,58)(51,71,59,68)(57,69,72,64);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,23)(13,24)(14,26)(15,27)(16,25)(18,38)(19,39)(20,41)(21,42)(22,40)(28,50)(29,51)(30,53)(31,52)(32,54)(33,56)(34,57)(35,58)(36,55)(37,59)(43,63)(44,64)(45,65)(46,66)(47,68)(48,67)(49,69)(60,70)(61,71)(62,72);(9,12,16,31)(10,18,22,45)(11,23,25,52)(13,29,15,37)(14,34,36,62)(17,38,40,65)(19,44,21,49)(20,47,48,61)(24,51,27,59)(26,57,55,72)(28,46,60,33)(30,32,43,35)(39,64,42,69)(41,68,67,71)(50,66,70,56)(53,54,63,58);(2,5)(6,8)(9,13,16,15)(10,19,22,21)(11,24,25,27)(12,29,31,37)(14,32,36,35)(17,39,40,42)(18,44,45,49)(20,46,48,33)(23,51,52,59)(26,54,55,58)(28,47,60,61)(30,34,43,62)(38,64,65,69)(41,66,67,56)(50,68,70,71)(53,57,63,72);(1,3,4,7)(2,6,5,8)(11,25)(17,40)(23,52)(24,27)(26,55)(38,65)(39,42)(41,67)(50,70)(51,59)(53,63)(54,58)(56,66)(57,72)(64,69)(68,71)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531791);