Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #381483 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14299 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,17,9,18,34,30,33)(4,19,23,42,24,10,22,37)(5,11,28,39,27,41,26,40)(15,35,20,31,46,43,51,56)(16,44,48,32,49,58,50,59)(21,52,54,60,53,36,25,38)(45,57,47,55,62,61,63,64);(1,3)(2,9)(4,20)(5,16)(6,30)(7,17)(8,18)(10,35)(11,32)(12,34)(13,33)(14,29)(15,22)(19,43)(21,47)(23,46)(24,51)(25,45)(26,50)(27,49)(28,48)(31,37)(36,57)(38,55)(39,58)(40,44)(41,59)(42,56)(52,61)(53,63)(54,62)(60,64);(1,4,8,24)(2,10,13,19)(3,15,18,46)(5,25,27,54)(6,23,7,22)(9,31,33,56)(11,38,41,60)(12,42,29,37)(14,43,34,35)(16,47,49,63)(17,51,30,20)(21,28,53,26)(32,57,59,61)(36,40,52,39)(44,55,58,64)(45,50,62,48);(1,5,7,28,8,27,6,26)(2,11,12,39,13,41,29,40)(3,16,17,48,18,49,30,50)(4,21,23,54,24,53,22,25)(9,32,34,58,33,59,14,44)(10,36,37,38,19,52,42,60)(15,45,20,47,46,62,51,63)(31,55,43,61,56,64,35,57)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288682);