Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461209 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16383 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,27,7,12,6,13)(3,14,17,40,18,9,28,32)(4,19,26,10,22,37,21,34)(5,23,29,36,25,11,24,52)(15,41,43,30,44,48,47,38)(16,31,45,57,46,39,54,59)(20,49,53,33,51,58,50,35)(42,60,61,55,62,64,63,56);(1,3,6,17,7,18,8,28)(2,9,13,32,12,14,27,40)(4,15,21,43,22,44,26,47)(5,16,24,45,25,46,29,54)(10,30,19,48,34,38,37,41)(11,31,36,57,23,39,52,59)(20,42,50,61,51,62,53,63)(33,55,49,64,35,56,58,60);(1,4)(2,10)(3,15)(5,20)(6,26)(7,22)(8,21)(9,30)(11,35)(12,34)(13,37)(14,38)(16,42)(17,47)(18,44)(19,27)(23,33)(24,53)(25,51)(28,43)(29,50)(31,56)(32,41)(36,49)(39,55)(40,48)(45,63)(46,62)(52,58)(54,61)(57,64)(59,60);(1,5,7,25)(2,11)(3,16,18,46)(4,20,22,51)(6,24,8,29)(9,31)(10,33)(12,23)(13,36)(14,39)(15,42,44,62)(17,45,28,54)(19,49)(21,50,26,53)(27,52)(30,55)(32,57)(34,35)(37,58)(38,56)(40,59)(41,60)(43,61,47,63)(48,64)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368406);