Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14187 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,26)(3,14,17,9,18,33,27,32)(4,10,21,36,22,37,28,35)(5,23,24,11,25,39,29,38)(15,40,44,30,19,48,43,56)(16,31,46,57,47,58,45,41)(20,50,53,59,52,54,51,34)(42,61,62,60,49,55,63,64);(1,3)(2,9)(4,19)(5,16)(6,27)(7,18)(8,17)(10,30)(11,31)(12,33)(13,32)(14,26)(15,22)(20,49)(21,43)(23,41)(24,46)(25,47)(28,44)(29,45)(34,55)(35,40)(36,48)(37,56)(38,58)(39,57)(42,52)(50,64)(51,62)(53,63)(54,60)(59,61);(1,4,8,21,7,22,6,28)(2,10,26,35,13,37,12,36)(3,15,27,43,18,19,17,44)(5,20,24,53,25,52,29,51)(9,30,33,48,32,56,14,40)(11,34,23,54,38,59,39,50)(16,42,45,63,47,49,46,62)(31,55,57,64,58,61,41,60);(1,5,8,24,7,25,6,29)(2,11,12,39,13,38,26,23)(3,16,17,46,18,47,27,45)(4,20,21,53,22,52,28,51)(9,31,33,57,32,58,14,41)(10,34,36,50,37,59,35,54)(15,42,44,62,19,49,43,63)(30,55,48,64,56,61,40,60)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284127);