Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #381470 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14299 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1345 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)(4,19,23,41)(5,11)(6,27)(7,12)(8,13)(9,17)(10,22,36,21)(15,34,46,30)(16,43)(18,32)(20,37,53,61)(24,35,54,62)(25,39)(26,40)(28,33)(29,38)(31,48)(42,45,52,51)(44,60,64,56)(47,57,55,63)(49,58)(50,59);(1,3)(2,9)(4,15)(5,16)(6,28)(7,17)(8,18)(10,34)(11,31)(12,32)(13,33)(14,27)(19,52)(20,44)(21,51)(22,45)(23,46)(24,55)(25,49)(26,50)(29,48)(30,36)(35,60)(37,57)(38,59)(39,43)(40,58)(41,42)(47,54)(53,64)(56,62)(61,63);(1,4)(2,10)(3,15)(5,24)(6,22)(7,21)(8,23)(9,30)(11,37)(12,41)(13,36)(14,42)(16,47)(17,51)(18,46)(19,27)(20,25)(26,54)(28,45)(29,53)(31,57)(32,52)(33,34)(35,38)(39,62)(40,61)(43,56)(44,48)(49,64)(50,55)(58,63)(59,60);(1,5,8,26)(2,11,13,40)(3,16,18,50)(4,20)(6,25,7,29)(9,31,33,58)(10,35)(12,38,27,39)(14,43,32,59)(15,44)(17,48,28,49)(19,37)(21,24)(22,54)(23,53)(30,56)(34,60)(36,62)(41,61)(42,63)(45,47)(46,64)(51,55)(52,57)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288667);