Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455116 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| 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 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(2,5)(6,8)(9,10)(11,17)(12,28)(13,34)(14,21)(15,20)(16,22)(18,31)(19,35)(23,43)(24,52)(25,41)(26,40)(27,42)(29,56)(30,59)(32,46)(33,45)(36,47)(37,48)(38,50)(39,53)(44,62)(49,69)(51,64)(54,65)(55,66)(57,60)(58,72)(61,68)(63,71)(67,70);(1,2,4,5)(3,6,7,8)(9,11,14,25,16,27,15,26)(10,17,21,41,22,42,20,40)(12,29,32,51,33,50,31,23)(13,24,35,53,36,54,37,55)(18,43,28,56,46,64,45,38)(19,39,47,65,48,66,34,52)(30,58,61,71,62,70,60,49)(44,67,57,69,59,72,68,63);(3,7)(6,8)(9,12,14,31,16,33,15,32)(10,18,20,28,22,46,21,45)(11,23,25,50,27,51,26,29)(13,30,35,60,36,62,37,61)(17,38,40,43,42,56,41,64)(19,44,34,57,48,59,47,68)(24,49,53,70,54,71,55,58)(39,63,52,67,66,69,65,72);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,30)(14,35)(15,37)(16,36)(17,39)(18,44)(20,34)(21,47)(22,48)(23,49)(25,53)(26,55)(27,54)(28,57)(29,58)(31,60)(32,61)(33,62)(38,63)(40,52)(41,65)(42,66)(43,67)(45,68)(46,59)(50,70)(51,71)(56,69)(64,72)Polycyclic presentation:
2^9|b^2=eg,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372403);