Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2856 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14836 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1171 conjugacy classes of subgroups; only the 244 normal subgroups are shown.
This group has 1171 conjugacy classes of subgroups — too many to draw. Only its 244 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,20,33,19,34)(4,21,25,10,26,37,24,38)(5,27,28,11,29,39,30,40)(16,31,22,35,45,57,44,41)(17,46,47,42,49,32,48,58)(23,52,54,50,55,36,53,60)(43,61,51,56,63,59,62,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,16,26,44,25,45)(5,17,28,47,29,49,30,48)(10,35,37,57,38,41,21,31)(11,32,27,42,40,46,39,58)(23,51,53,43,55,62,54,63)(36,59,60,64,52,61,50,56);(1,4,6,25,8,26,7,24)(2,10,12,21,14,38,13,37)(3,16,19,44,20,45,18,22)(5,23,28,54,29,55,30,53)(9,31,33,35,34,57,15,41)(11,36,27,50,40,52,39,60)(17,43,48,62,49,63,47,51)(32,56,58,59,46,64,42,61);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,39)(14,40)(15,42)(16,43)(18,47)(19,48)(20,49)(21,50)(22,51)(24,53)(25,54)(26,55)(31,56)(33,58)(34,46)(35,59)(37,60)(38,52)(41,61)(44,62)(45,63)(57,64)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309394);