Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219518 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,18,12,17,10,16)(7,19,22,36,23,45,33,13)(8,24,27,39,28,14,26,40)(9,15,31,44,32,43,30,42)(20,48,50,37,51,46,55,34)(21,47,53,63,54,66,60,35)(25,56,57,41,58,38,29,59)(49,62,52,64,67,65,68,61);(1,2)(3,4)(5,7)(6,13,17,36)(8,20)(9,21)(10,22)(11,33)(12,23)(14,37,24,34)(15,35,43,63)(16,19,18,45)(25,49)(26,50)(27,55)(28,51)(29,52)(30,53)(31,60)(32,54)(38,64,56,61)(39,46,40,48)(41,65,59,62)(42,47,44,66)(57,68)(58,67);(2,4)(5,8,12,28)(6,14,17,24)(7,20,23,51)(9,29,32,57)(10,27,11,26)(13,34,36,37)(15,41,43,59)(16,40,18,39)(19,46,45,48)(21,52,54,68)(22,55,33,50)(25,31,58,30)(35,62,63,65)(38,44,56,42)(47,61,66,64)(49,60,67,53);(5,9,11,31,12,32,10,30)(6,15,18,44,17,43,16,42)(7,21,33,60,23,54,22,53)(8,25,27,57,28,58,26,29)(13,35,45,66,36,63,19,47)(14,38,40,59,24,56,39,41)(20,49,55,68,51,67,50,52)(34,61,46,65,37,64,48,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287383);