Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624842 |
| 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 | |
1359 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)(9,10,13,18)(11,21,23,16)(12,26,28,57)(14,33,31,40)(15,35,32,20)(17,39,41,29)(19,27,43,34)(22,47,49,56)(24,54,52,64)(25,38,53,46)(30,63,61,59)(36,50,44,66)(37,55,45,48)(42,62,58,60)(51,72,70,68)(65,69,67,71);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,36)(18,21)(19,37)(20,38)(26,56)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,64)(34,55)(35,46)(39,66)(40,54)(41,44)(42,65)(43,45)(47,57)(58,67)(59,72)(60,69)(61,70)(62,71)(63,68);(9,12,27,60,15,30,14,29,13,28,34,62,32,61,31,39)(10,17,40,63,20,42,19,26,18,41,33,59,35,58,43,57)(11,22,48,69,25,51,24,50,23,49,55,71,53,70,52,66)(16,36,54,68,38,65,37,56,21,44,64,72,46,67,45,47);(2,5)(6,8)(11,23)(12,28)(16,21)(17,41)(24,52)(25,53)(26,57)(29,39)(30,61)(37,45)(38,46)(42,58)(48,55)(54,64)(59,63)(60,62);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(17,41)(19,43)(20,35)(22,49)(24,52)(25,53)(26,57)(33,40)(42,58)(48,55)(50,66)(51,70)(59,63)(69,71)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,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=df,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532201);