Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150723 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11839 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #381 | C23 |
| prime radical | #23987 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,8,18,22)(3,11,13,27,14,24,30,32)(4,15,17,31,16,9,23,25)(7,19,21,12,20,28,26,29);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,24)(10,26)(11,15)(16,29)(17,19)(18,30)(21,22)(23,28)(25,32)(27,31);(2,8)(4,16)(5,18)(7,20)(11,24)(13,30)(19,28)(25,31)Polycyclic presentation:
2^9|a^2=e,d^2=g,e^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efg,h^b=hi,d^c=dhi,e^c=ei,e^d=eh,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72237478, #72237867, #72237968 |
| maximal normal subgroup of | #72237478, #72237867, #72237968 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57921);