Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138556 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19128 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,5,8)(3,6)(4,9,13,14)(7,12)(10,16)(11,15)(18,22,24,26)(19,23)(20,28)(25,30,32,29)(27,31);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,18,23,30,28,32,19,26,21,24,31,29,20,25,27,22);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(18,24)(22,26)(25,32)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=ehi,f^c=fhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287091, #72287109, #72287115, … |
| maximal normal subgroup of | #72287091, #72287109, #72287115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45754);