Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455572 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,23)(12,18)(13,34,38,50)(14,21)(15,20)(16,22)(17,26)(19,37,49,36)(24,43)(25,55,58,40)(27,42)(28,41)(29,39)(30,48,61,69)(31,46)(32,45)(33,47)(35,44,60,68)(51,57,62,59)(52,64,63,70)(53,66)(54,65)(56,72,71,67);(1,3,5,4)(2,6,7,8)(9,11,14,26,16,28,15,27)(10,17,20,23,22,42,21,41)(12,29,31,24,33,53,32,54)(13,25,37,59,38,58,36,57)(18,43,45,39,47,65,46,66)(19,40,50,51,49,55,34,62)(30,56,60,52,61,71,35,63)(44,67,69,64,68,72,48,70);(2,7)(6,8)(9,12,16,33)(10,18,22,47)(11,24,28,54)(13,35,38,60)(14,32,15,31)(17,39,42,66)(19,48,49,69)(20,46,21,45)(23,43,41,65)(25,56,58,71)(26,29,27,53)(30,37,61,36)(34,44,50,68)(40,67,55,72)(51,70,62,64)(52,59,63,57);(1,4,5,3)(2,8,7,6)(9,13)(10,19)(11,25)(12,30)(14,37)(15,36)(16,38)(17,40)(18,44)(20,50)(21,34)(22,49)(23,51)(24,52)(26,59)(27,57)(28,58)(29,56)(31,60)(32,35)(33,61)(39,64)(41,62)(42,55)(43,67)(45,69)(46,48)(47,68)(53,71)(54,63)(65,72)(66,70)Polycyclic presentation:
2^9|a^2=efg,b^2=fgi,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374943);