Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14005 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,29)(3,14,19,32,18,39,17,9)(4,10,23,34,24,35,22,53)(5,25,28,36,27,54,26,11)(15,40,43,30,20,37,42,56)(16,44,47,59,46,31,45,38)(21,49,51,57,52,64,50,33)(41,55,62,58,48,63,61,60);(1,3,6,17,7,18,8,19)(2,9,12,14,13,32,29,39)(4,20,23,42,24,15,22,43)(5,16,26,45,27,46,28,47)(10,30,53,40,35,56,34,37)(11,31,25,38,36,44,54,59)(21,48,51,61,52,41,50,62)(33,55,64,60,57,63,49,58);(1,4)(2,10)(3,15,18,20)(5,21)(6,22)(7,24)(8,23)(9,30,32,56)(11,33)(12,34)(13,35)(14,37,39,40)(16,41,46,48)(17,42,19,43)(25,49)(26,50)(27,52)(28,51)(29,53)(31,55,44,63)(36,57)(38,58,59,60)(45,61,47,62)(54,64);(1,5,6,26,7,27,8,28)(2,11,12,25,13,36,29,54)(3,16,17,45,18,46,19,47)(4,21,22,50,24,52,23,51)(9,31,14,38,32,44,39,59)(10,33,34,49,35,57,53,64)(15,41,42,61,20,48,43,62)(30,55,37,58,56,63,40,60)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274746);