Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1348 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)(23,52)(24,39)(25,47)(26,40)(27,41)(28,57)(29,43)(30,48)(31,44)(33,46)(36,45)(38,54)(42,59)(49,69)(50,66)(51,64)(53,61)(55,65)(56,63)(58,62)(60,68)(67,70)(71,72);(2,5)(6,8)(9,11,16,26)(10,17,21,40)(12,27,31,50)(13,23,33,53)(14,25,15,24)(18,41,44,66)(19,38,45,65)(20,47,35,39)(22,30,51,29)(28,56,58,71)(32,61,46,52)(34,55,36,54)(37,48,64,43)(42,67,68,69)(49,60,70,59)(57,72,62,63);(9,12,16,31)(10,18,21,44)(11,22,26,51)(13,28,33,58)(14,30,15,29)(17,37,40,64)(19,42,45,68)(20,48,35,43)(23,49,53,70)(24,50,25,27)(32,62,46,57)(34,60,36,59)(38,63,65,72)(39,66,47,41)(52,67,61,69)(54,71,55,56);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,23)(12,28)(14,34)(15,36)(16,33)(17,38)(18,42)(20,46)(21,45)(22,49)(24,54)(25,55)(26,53)(27,56)(29,59)(30,60)(31,58)(32,35)(37,63)(39,61)(40,65)(41,67)(43,62)(44,68)(47,52)(48,57)(50,71)(51,70)(64,72)(66,69)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307197);