Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| 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 | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1212 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)(4,10,22,37,23,38,21,56)(5,24,26,11,28,40,27,41)(15,43,19,49,45,35,51,31)(16,46,48,42,47,32,25,39)(20,52,53,36,55,60,54,61)(44,58,50,62,63,64,57,59);(1,3)(2,9)(4,19)(5,25)(6,30)(7,17)(8,18)(10,35)(11,39)(12,34)(13,33)(14,29)(15,21)(16,26)(20,44)(22,45)(23,51)(24,32)(27,47)(28,48)(31,37)(36,58)(38,43)(40,46)(41,42)(49,56)(50,53)(52,59)(54,57)(55,63)(60,62)(61,64);(1,4,7,22,8,23,6,21)(2,10,12,37,13,38,29,56)(3,15,30,51,18,45,17,19)(5,20,26,53,28,55,27,54)(9,31,14,35,33,49,34,43)(11,36,40,60,41,61,24,52)(16,44,25,57,47,63,48,50)(32,58,42,59,46,64,39,62);(1,5)(2,11)(3,16)(4,20)(6,27)(7,26)(8,28)(9,32)(10,36)(12,40)(13,41)(14,42)(15,44)(17,48)(18,47)(19,50)(21,54)(22,53)(23,55)(24,29)(25,30)(31,58)(33,46)(34,39)(35,59)(37,60)(38,61)(43,62)(45,63)(49,64)(51,57)(52,56)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,g^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^b=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296225);