Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| 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 | |
1548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6)(7,12)(8,13)(9,25,26,14)(10,15)(11,16)(17,31)(18,45,46,32)(19,33)(20,34)(21,35)(22,36,50,52)(23,37)(24,38)(27,53,54,39)(28,40)(29,41)(30,42)(43,57,65,66)(44,58)(47,63,55,59)(48,60)(49,61,67,68)(51,62,56,64);(1,2)(3,4)(5,7,11,20)(6,12,16,34)(8,21,24,48)(9,18,26,46)(10,28,30,19)(13,35,38,60)(14,32,25,45)(15,40,42,33)(17,29,44,23)(22,49,50,67)(27,55,54,47)(31,41,58,37)(36,61,52,68)(39,63,53,59)(43,56,65,51)(57,64,66,62);(5,8,11,24)(6,13,16,38)(7,17,20,44)(9,22,26,50)(10,29,30,23)(12,31,34,58)(14,36,25,52)(15,41,42,37)(18,43,46,65)(19,48,28,21)(27,56,54,51)(32,57,45,66)(33,60,40,35)(39,64,53,62)(47,67,55,49)(59,68,63,61);(1,3,4,2)(5,9)(6,14)(7,18)(8,22)(10,27)(11,26)(12,32)(13,36)(15,39)(16,25)(17,43)(19,47)(20,46)(21,49)(23,51)(24,50)(28,55)(29,56)(30,54)(31,57)(33,59)(34,45)(35,61)(37,62)(38,52)(40,63)(41,64)(42,53)(44,65)(48,67)(58,66)(60,68)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326918);