Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | #1459 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
311 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,8,12,6)(5,9)(10,15)(11,14,16,13)(17,18,20,24,21,25,29,32,22,26,19,23,27,30,28,31);(2,6)(3,10)(4,12)(7,14)(8,9)(13,15);(1,3,5,10)(2,7,9,15)(4,11,12,16)(6,13,8,14)(17,19,21,28,22,20,27,29)(18,23,25,31,26,24,30,32)Polycyclic presentation:
2^9|a^2=e,c^2=gh,d^2=h,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dh,e^b=efh,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29575);