Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384259 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2333 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14481 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,19,9,20,34,18,33)(4,21,25,37,24,10,23,38)(5,26,28,11,29,40,30,41)(16,42,45,31,46,59,44,35)(17,47,27,57,48,32,49,39)(22,51,53,50,54,56,55,36)(43,60,52,61,63,58,64,62);(1,3)(2,9)(4,16)(5,27)(6,19)(7,20)(8,18)(10,35)(11,39)(12,34)(13,33)(14,15)(17,28)(21,31)(22,52)(23,45)(24,46)(25,44)(26,47)(29,49)(30,48)(32,40)(36,61)(37,42)(38,59)(41,57)(43,53)(50,62)(51,60)(54,64)(55,63)(56,58);(1,4,8,25,7,24,6,23)(2,10,14,38,13,21,12,37)(3,16,18,44,20,46,19,45)(5,22,30,55,29,54,28,53)(9,31,15,42,33,35,34,59)(11,36,26,56,41,50,40,51)(17,43,27,52,48,63,49,64)(32,58,39,62,47,60,57,61);(1,5,6,28,7,29,8,30)(2,11,14,26,13,41,12,40)(3,17,18,27,20,48,19,49)(4,22,23,53,24,54,25,55)(9,32,34,57,33,47,15,39)(10,36,38,56,21,50,37,51)(16,43,44,52,46,63,45,64)(31,58,59,61,35,60,42,62)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294770);