Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14267 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
934 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)(3,14,19,33)(4,20,25,42)(5,11)(6,12)(7,29)(8,13)(9,17,34,18)(10,23,37,24)(15,45)(16,44,52,60)(21,35)(22,38,54,62)(26,56,55,36)(27,39)(28,41)(30,40)(31,48)(32,50,61,51)(43,47)(46,64)(49,63)(53,59)(57,58);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,35)(11,32)(12,33)(13,34)(14,29)(15,24)(20,31)(22,53)(23,48)(25,47)(26,57)(27,51)(28,52)(30,50)(36,59)(37,43)(38,58)(39,60)(40,44)(41,61)(42,45)(46,55)(49,54)(56,63)(62,64);(1,4)(2,10)(3,15)(5,26)(6,24)(7,23)(8,25)(9,31)(11,38)(12,42)(13,37)(14,43)(16,49)(17,21)(18,47)(19,48)(20,29)(22,27)(28,55)(30,54)(32,59)(33,35)(34,45)(36,39)(40,56)(41,62)(44,58)(46,50)(51,57)(52,53)(60,64)(61,63);(1,5,6,27,8,28,7,30)(2,11,12,39,13,41,29,40)(3,16,18,51,19,52,17,50)(4,22,23,26,25,54,24,55)(9,32,33,60,34,61,14,44)(10,36,20,38,37,56,42,62)(15,46,21,53,48,57,47,49)(31,58,43,63,45,64,35,59)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287625);