Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2352 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13885 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,14)(3,15,19,33,20,40,18,9)(4,21,24,36,26,10,25,37)(5,27,29,38,30,39,28,11)(16,34,22,51,44,41,49,31)(17,45,47,60,48,32,46,42)(23,53,54,50,56,57,55,35)(43,58,52,59,62,63,64,61);(1,3)(2,9,13,33)(4,22)(5,17)(6,18)(7,19)(8,20)(10,34,21,41)(11,32,38,45)(12,15,14,40)(16,25)(23,52)(24,44)(26,49)(27,42,39,60)(28,46)(29,47)(30,48)(31,37,51,36)(35,59,50,61)(43,55)(53,63,57,58)(54,62)(56,64);(1,4,6,24,8,26,7,25)(2,10,14,37,13,21,12,36)(3,16,18,22,20,44,19,49)(5,23,28,54,30,56,29,55)(9,31,40,34,33,51,15,41)(11,35,39,53,38,50,27,57)(17,43,46,52,48,62,47,64)(32,58,60,59,45,63,42,61);(1,5,6,28,8,30,7,29)(2,11,12,27,13,38,14,39)(3,17,18,46,20,48,19,47)(4,23,24,54,26,56,25,55)(9,32,15,42,33,45,40,60)(10,35,36,57,21,50,37,53)(16,43,22,52,44,62,49,64)(31,58,41,61,51,63,34,59)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268169);