Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
1261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,10,16,12,17,11,33)(7,18,21,38,22,37,32,13)(8,14,25,41,27,43,26,42)(9,28,30,15,31,45,34,46)(19,47,49,62,50,39,23,35)(20,51,29,59,53,36,52,44)(24,55,56,40,58,64,57,65)(48,67,60,63,68,61,54,66);(5,7,12,22)(6,13,17,38)(8,23,27,49)(9,29,31,52)(10,32,11,21)(14,39,43,47)(15,44,46,59)(16,37,33,18)(19,25,50,26)(20,34,53,30)(24,48,58,68)(28,51,45,36)(35,41,62,42)(40,61,65,67)(54,57,60,56)(55,66,64,63);(1,2,3,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,27)(13,35)(15,40)(16,41)(17,43)(18,47)(20,48)(21,49)(22,50)(23,32)(28,55)(29,60)(30,56)(31,58)(33,42)(34,57)(36,61)(37,39)(38,62)(44,66)(45,64)(46,65)(51,67)(52,54)(53,68)(59,63);(5,9,11,34,12,31,10,30)(6,15,33,28,17,46,16,45)(7,20,32,52,22,53,21,29)(8,24,26,57,27,58,25,56)(13,36,37,59,38,51,18,44)(14,40,42,55,43,65,41,64)(19,48,23,54,50,68,49,60)(35,61,39,63,62,67,47,66)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=f,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331030);