Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455155 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,18)(13,32)(14,20)(15,35)(16,21)(19,34)(22,37)(23,51)(24,39)(25,54)(26,40)(27,41)(28,57)(29,43)(30,47)(31,44)(33,46)(36,45)(38,53)(42,59)(48,69)(49,66)(50,63)(52,65)(55,64)(56,62)(58,61)(60,68)(67,70)(71,72);(2,5)(6,8)(9,11,14,24,16,26,15,25)(10,17,20,39,21,40,35,54)(12,27,29,50,31,49,30,22)(13,23,34,53,33,52,36,55)(18,41,43,63,44,66,47,37)(19,38,46,65,45,64,32,51)(28,56,59,70,58,71,60,48)(42,67,61,72,68,69,57,62);(1,3)(2,6)(4,7)(5,8)(9,12,16,31)(10,18,21,44)(11,22,26,50)(13,28,33,58)(14,30,15,29)(17,37,40,63)(19,42,45,68)(20,47,35,43)(23,48,52,70)(24,49,25,27)(32,61,46,57)(34,60,36,59)(38,62,64,72)(39,66,54,41)(51,67,65,69)(53,71,55,56);(3,7)(6,8)(9,13)(10,19)(11,23)(12,28)(14,34)(15,36)(16,33)(17,38)(18,42)(20,46)(21,45)(22,48)(24,53)(25,55)(26,52)(27,56)(29,59)(30,60)(31,58)(32,35)(37,62)(39,65)(40,64)(41,67)(43,61)(44,68)(47,57)(49,71)(50,70)(51,54)(63,72)(66,69)Polycyclic presentation:
2^9|b^2=g,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371206);