Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| 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 | |
999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,17,9,18,35,19,34)(4,20,23,38,24,10,31,39)(5,25,27,11,29,41,28,42)(15,36,45,32,46,50,21,52)(16,47,49,43,48,33,26,40)(22,54,57,51,56,58,55,37)(44,61,63,60,64,59,53,62);(1,3)(2,9)(4,21)(5,26)(6,19)(7,17)(8,18)(10,36)(11,40)(12,35)(13,34)(14,30)(15,23)(16,27)(20,50)(22,44)(24,45)(25,33)(28,48)(29,49)(31,46)(32,39)(37,61)(38,52)(41,47)(42,43)(51,59)(53,55)(54,60)(56,64)(57,63)(58,62);(1,4)(2,10,13,20)(3,15,18,46)(5,22)(6,31)(7,23)(8,24)(9,32)(11,37,42,51)(12,39,30,38)(14,36)(16,44,48,64)(17,45,19,21)(25,58,41,54)(26,53,49,63)(27,57)(28,55)(29,56)(33,59)(34,52)(35,50)(40,62)(43,60)(47,61);(1,5)(2,11)(3,16)(4,22)(6,28)(7,27)(8,29)(9,33)(10,37)(12,41)(13,42)(14,43)(15,44)(17,49)(18,48)(19,26)(20,51)(21,53)(23,57)(24,56)(25,30)(31,55)(32,59)(34,47)(35,40)(36,60)(38,58)(39,54)(45,63)(46,64)(50,62)(52,61)Polycyclic presentation:
2^9|a^2=h,c^2=efg,e^2=i,g^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,g^b=gi,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, 293579);