Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93205 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1343 |
| commutator | C2×C22⋊C8 | C2×C4 |
| Frattini, cosocle | #1818 | C22 |
| prime radical | #3186 | C4 |
| LCS | G ▷ C2×C22⋊C8 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1818 ◁ G | |
| LECS | G ▷ #1818 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
176 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 128 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,9)(3,8,14,25)(5,10,19,27)(6,11,18,26)(7,20,21,12)(13,24,16,29)(15,22,17,30)(23,31,32,28)(33,34,37,41,38,42,50,58)(35,43,45,59,46,55,62,64)(36,47,48,63,49,40,54,56)(39,51,52,44,53,60,57,61);(2,7)(3,12)(4,16)(5,13)(8,10)(9,25)(11,23)(15,28)(18,19)(20,31)(21,29)(22,24)(26,30)(27,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,55)(41,57)(42,53)(43,47)(48,51)(49,61)(50,62)(52,58)(54,60)(56,64)(59,63)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gh,c^b=cf,d^b=def,f^b=fh,g^b=gi,d^c=dg,e^c=ei,e^d=eh35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405);