Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| 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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1280 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)(11,17)(12,27)(13,19)(14,33)(15,36)(16,21)(18,31)(20,34)(22,45)(23,38)(24,39)(25,52)(26,54)(28,44)(29,55)(30,56)(32,61)(35,41)(37,48)(40,53)(42,58)(43,59)(46,65)(47,66)(49,72)(50,68)(51,62)(57,60)(63,69)(64,71)(67,70);(1,3)(2,5)(4,8)(6,7)(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)(30,50)(31,48)(32,49)(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);(9,12,28,58,15,32,14,31,16,30,34,60,13,29,35,59)(10,18,41,61,19,42,20,27,21,43,33,55,36,57,44,56)(11,22,46,69,26,49,25,48,24,50,53,70,23,47,51,71)(17,37,62,72,38,63,40,45,39,64,52,66,54,67,65,68);(3,8)(5,7)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,29,30,32)(14,28,35,34)(17,38,39,54)(18,42,43,57)(20,33,44,41)(22,47,50,49)(25,46,51,53)(27,55,56,61)(31,58,59,60)(37,63,64,67)(40,52,65,62)(45,66,68,72)(48,69,71,70);(2,6)(5,7)(11,24)(12,30)(17,39)(18,43)(23,26)(25,51)(27,56)(29,32)(31,59)(38,54)(40,65)(42,57)(46,53)(52,62)(55,61)(58,60)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,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=ei,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532106);