Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #365053 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13973 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,19,32,18,41,17,9)(4,10,22,36,23,37,29,35)(5,24,27,38,26,54,25,11)(15,39,43,30,20,49,48,33)(16,44,47,61,46,31,45,40)(21,51,52,34,53,58,55,59)(42,57,62,60,50,56,63,64);(1,3,6,17,7,18,8,19)(2,9,12,14,13,32,28,41)(4,20,29,43,23,15,22,48)(5,16,25,45,26,46,27,47)(10,33,36,39,37,30,35,49)(11,31,24,40,38,44,54,61)(21,50,55,62,53,42,52,63)(34,57,58,60,59,56,51,64);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,29)(9,30)(11,34)(12,35)(13,37)(14,39)(16,42)(17,43)(18,20)(19,48)(24,51)(25,52)(26,53)(27,55)(28,36)(31,56)(32,33)(38,59)(40,60)(41,49)(44,57)(45,62)(46,50)(47,63)(54,58)(61,64);(1,5,6,25,7,26,8,27)(2,11,12,24,13,38,28,54)(3,16,17,45,18,46,19,47)(4,21,22,52,23,53,29,55)(9,31,14,40,32,44,41,61)(10,34,35,51,37,59,36,58)(15,42,43,62,20,50,48,63)(30,56,39,60,33,57,49,64)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=e,e^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,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272447);