Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14427 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,17,9,18,35,29,34)(4,19,22,38,23,10,30,39)(5,24,26,11,27,41,31,42)(15,43,47,32,20,52,46,36)(16,48,50,44,49,33,25,40)(21,54,58,51,57,59,56,37)(45,61,64,63,53,60,55,62);(1,3)(2,9)(4,20)(5,25)(6,29)(7,18)(8,17)(10,36)(11,40)(12,35)(13,34)(14,28)(15,23)(16,26)(19,32)(21,55)(22,46)(24,33)(27,50)(30,47)(31,49)(37,62)(38,52)(39,43)(41,48)(42,44)(45,58)(51,63)(53,56)(54,61)(57,64)(59,60);(1,4,8,22,7,23,6,30)(2,10,28,38,13,19,12,39)(3,15,17,47,18,20,29,46)(5,21,26,58,27,57,31,56)(9,32,14,43,34,36,35,52)(11,37,24,59,42,51,41,54)(16,45,50,64,49,53,25,55)(33,60,44,63,48,61,40,62);(1,5)(2,11)(3,16)(4,21)(6,31)(7,27)(8,26)(9,33)(10,37)(12,41)(13,42)(14,44)(15,45)(17,50)(18,49)(19,51)(20,53)(22,58)(23,57)(24,28)(25,29)(30,56)(32,60)(34,48)(35,40)(36,61)(38,59)(39,54)(43,63)(46,55)(47,64)(52,62)Polycyclic presentation:
2^9|a^2=h,c^2=efg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293576);