Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×#97 | C2×C4 |
| Frattini, cosocle | #1985 | C22 |
| socle | C22 | #1343 |
| prime radical | C23 | #327 |
| LCS | G ▷ C2×#97 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1985 ◁ G | |
| LECS | G ▷ #1985 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1 | |
162 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 256 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,9,6,11,18,28)(3,12,13,26,15,23,30,31)(5,10,20,27,21,8,17,25)(7,16,22,19,24,29,32,14)(33,34,37,42,39,44,38,43)(35,45,47,61,49,62,48,58)(36,50,52,59,53,60,54,41)(40,51,55,46,57,63,56,64);(1,3,6,15)(2,7,11,24)(4,16,18,29)(5,19,21,14)(8,23,10,12)(9,26,28,31)(13,17,30,20)(22,27,32,25)(33,35)(34,40)(36,51)(37,48)(38,47)(39,49)(41,58)(42,56)(43,55)(44,57)(45,50)(46,52)(53,63)(54,64)(59,61)(60,62)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fi,d^2=hi,e^2=hi,f^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=defhi,e^b=ehi,f^b=fi,g^b=gi,d^c=dghi,e^c=ei,e^d=eh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 432);