Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392727 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14694 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3597 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)(4,19,22,10)(5,23,26,38)(6,27)(7,12)(8,13)(9,17)(11,25,37,24)(15,39,43,58)(16,45,47,31)(18,32)(20,51)(21,35,28,36)(29,33)(30,42,49,44)(34,52)(40,46,55,48)(41,60)(50,54)(53,56)(57,62)(59,61)(63,64);(1,3,6,17,8,18,7,29)(2,9,12,14,13,33,27,32)(4,15,28,44,22,43,21,42)(5,16,25,48,26,47,24,46)(10,30,36,39,19,49,35,58)(11,31,38,40,37,45,23,55)(20,41,52,61,53,62,54,63)(34,57,51,59,50,60,56,64);(1,4)(2,10)(3,15)(5,20)(6,28)(7,21)(8,22)(9,30)(11,34)(12,36)(13,19)(14,39)(16,41)(17,44)(18,43)(23,56)(24,54)(25,52)(26,53)(27,35)(29,42)(31,57)(32,58)(33,49)(37,50)(38,51)(40,59)(45,60)(46,63)(47,62)(48,61)(55,64);(1,5)(2,11)(3,16)(4,20)(6,25)(7,24)(8,26)(9,31)(10,34)(12,38)(13,37)(14,40)(15,41)(17,48)(18,47)(19,50)(21,54)(22,53)(23,27)(28,52)(29,46)(30,57)(32,55)(33,45)(35,56)(36,51)(39,59)(42,63)(43,62)(44,61)(49,60)(58,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299923);