Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(2,6,8,11)(3,9,10,12)(18,24)(19,25,28,32)(20,22)(21,26)(27,31,29,30);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,18,22,21,23,26,20,24)(19,25,27,30,28,32,29,31);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,19,23,28)(18,25,26,32)(20,29,22,27)(21,30,24,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=i,d^2=gi,e^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287279, #72287346, #72287363, … |
| maximal normal subgroup of | #72287279, #72287346, #72287363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44696);