Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384166 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14509 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
751 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)(3,13,17,38)(4,18,21,9)(5,22)(6,11)(7,12)(8,16,30,26)(10,24)(14,39)(15,43,44,29)(19,47,50,53)(20,46,52,33)(23,54,55,34)(25,36)(27,35)(28,51)(31,41)(32,49,45,63)(37,56)(40,58)(42,59)(48,60)(57,62)(61,64);(1,3)(2,8)(4,14)(5,23)(6,26)(7,17)(9,31)(10,34)(11,38)(12,30)(13,35)(15,24)(16,27)(18,28)(19,48)(20,51)(21,42)(22,43)(25,55)(29,36)(32,60)(33,39)(37,54)(40,49)(41,52)(44,56)(45,61)(46,59)(47,58)(50,64)(53,57)(62,63);(1,4,6,20,7,21,27,52)(2,9,11,33,12,18,35,46)(3,14,26,51,17,42,16,41)(5,19,24,49,25,50,56,63)(8,28,38,59,30,31,13,39)(10,32,36,47,37,45,22,53)(15,40,55,64,44,62,23,48)(29,57,54,60,43,58,34,61);(1,5,6,24,7,25,27,56)(2,10,35,22,12,37,11,36)(3,15,16,23,17,44,26,55)(4,19,20,49,21,50,52,63)(8,29,38,54,30,43,13,34)(9,32,46,53,18,45,33,47)(14,40,41,48,42,62,51,64)(28,57,59,60,31,58,39,61)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295222);