Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,23,26,38)(16,20)(17,30,32,21)(18,22)(19,25)(24,39,27,41)(28,43,44,36)(29,37)(31,34)(33,40)(35,42);(13,15,18,26)(14,19,22,34)(16,27,29,24)(17,31,32,25)(20,35,37,33)(21,38,30,23)(28,40,44,42)(36,39,43,41);(2,4)(13,16,18,29)(14,20,22,37)(15,24,26,27)(17,28,32,44)(19,33,34,35)(21,36,30,43)(23,39,38,41)(25,40,31,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,f^b=fgi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36416);