Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388466 |
| 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 | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1133 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)(3,13)(4,9,21,35)(5,22,25,37)(6,26)(7,12)(8,16)(10,24,38,28)(11,29)(14,40,18,50)(15,45,46,31)(17,42)(19,52)(20,39,27,34)(23,36,48,41)(30,44,58,47)(32,49)(33,54)(43,63)(51,57)(53,62)(55,61)(56,60)(59,64);(1,3,6,16,7,17,29,49)(2,8,11,13,12,32,26,42)(4,18,20,47,21,14,27,44)(5,23,24,46,25,48,28,15)(9,30,34,50,35,58,39,40)(10,36,37,45,38,41,22,31)(19,53,54,43,55,64,56,51)(33,59,52,63,60,62,61,57);(1,4,6,27,7,21,29,20)(2,9,11,39,12,35,26,34)(3,14,16,47,17,18,49,44)(5,19,28,54,25,55,24,56)(8,30,13,40,32,58,42,50)(10,33,22,52,38,60,37,61)(15,43,48,64,46,51,23,53)(31,57,41,59,45,63,36,62);(1,5)(2,10)(3,15)(4,19)(6,28)(7,25)(8,31)(9,33)(11,22)(12,38)(13,41)(14,43)(16,48)(17,46)(18,51)(20,56)(21,55)(23,49)(24,29)(26,37)(27,54)(30,57)(32,45)(34,61)(35,60)(36,42)(39,52)(40,59)(44,53)(47,64)(50,62)(58,63)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 295662);