Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14533 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,17,9,18,34,29,33)(4,10,23,36,22,37,21,55)(5,24,26,11,27,40,30,39)(15,43,45,59,19,41,50,31)(16,46,48,42,47,32,25,38)(20,51,53,35,54,61,56,62)(44,58,52,60,49,64,57,63);(1,3)(2,9)(4,19)(5,25)(6,29)(7,17)(8,18)(10,31)(11,38)(12,34)(13,33)(14,28)(15,22)(16,26)(20,52)(21,45)(23,50)(24,32)(27,48)(30,47)(35,60)(36,43)(37,59)(39,42)(40,46)(41,55)(44,56)(49,53)(51,58)(54,57)(61,64)(62,63);(1,4,7,23,8,22,6,21)(2,10,12,36,13,37,28,55)(3,15,29,50,18,19,17,45)(5,20,26,53,27,54,30,56)(9,31,14,41,33,59,34,43)(11,35,40,61,39,62,24,51)(16,44,25,57,47,49,48,52)(32,58,42,63,46,64,38,60);(1,5)(2,11)(3,16)(4,20)(6,30)(7,26)(8,27)(9,32)(10,35)(12,40)(13,39)(14,42)(15,44)(17,48)(18,47)(19,49)(21,56)(22,54)(23,53)(24,28)(25,29)(31,58)(33,46)(34,38)(36,61)(37,62)(41,63)(43,60)(45,52)(50,57)(51,55)(59,64)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296222);