Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| 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 | |
1375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10)(11,17)(12,27)(13,19)(14,33)(15,36)(16,21)(18,30)(20,34)(22,45)(23,38)(24,39)(25,52)(26,54)(28,44)(29,55)(31,61)(32,57)(35,41)(37,49)(40,53)(42,58)(43,60)(46,65)(47,66)(48,67)(50,72)(51,62)(56,59)(63,69)(64,70)(68,71);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,22)(13,23)(14,25)(15,26)(16,24)(18,37)(19,38)(20,40)(21,39)(27,45)(28,46)(29,47)(30,49)(31,50)(32,48)(33,52)(34,53)(35,51)(36,54)(41,62)(42,63)(43,64)(44,65)(55,66)(56,68)(57,67)(58,69)(59,71)(60,70)(61,72);(9,12,28,58,15,31,14,30,16,32,34,59,13,29,35,60)(10,18,41,61,19,42,20,27,21,43,33,55,36,56,44,57)(11,22,46,69,26,50,25,49,24,48,53,71,23,47,51,70)(17,37,62,72,38,63,40,45,39,64,52,66,54,68,65,67);(2,5)(6,8)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,29,32,31)(14,28,35,34)(17,38,39,54)(18,42,43,56)(20,33,44,41)(22,47,48,50)(25,46,51,53)(27,55,57,61)(30,58,60,59)(37,63,64,68)(40,52,65,62)(45,66,67,72)(49,69,70,71);(1,3,4,7)(2,6,5,8)(11,24)(17,39)(22,48)(23,26)(25,51)(37,64)(38,54)(40,65)(45,67)(46,53)(47,50)(49,70)(52,62)(63,68)(66,72)(69,71)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532189);