Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14520 |
| 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 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,20,34,19,33,18,9)(4,21,25,36,26,41,24,10)(5,11,30,38,29,40,28,39)(16,31,46,49,22,51,45,42)(17,43,47,32,48,37,27,58)(23,53,56,61,57,50,55,35)(44,63,54,62,52,59,64,60);(1,3,6,18,8,19,7,20)(2,9,12,15,14,34,13,33)(4,22,24,46,26,16,25,45)(5,27,28,48,29,47,30,17)(10,31,21,49,36,51,41,42)(11,37,38,58,40,43,39,32)(23,54,55,44,57,64,56,52)(35,60,53,63,61,62,50,59);(1,4)(2,10,14,36)(3,16)(5,23)(6,25)(7,24)(8,26)(9,31,34,51)(11,35,40,61)(12,41,13,21)(15,42,33,49)(17,44)(18,46)(19,22)(20,45)(27,54)(28,56)(29,57)(30,55)(32,59,58,63)(37,62,43,60)(38,50,39,53)(47,64)(48,52);(1,5,8,29)(2,11,14,40)(3,17,19,48)(4,23,26,57)(6,30,7,28)(9,32,34,58)(10,35,36,61)(12,39,13,38)(15,43,33,37)(16,44,22,52)(18,47,20,27)(21,50,41,53)(24,56,25,55)(31,59,51,63)(42,60,49,62)(45,54,46,64)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295655);