Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#2813 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
963 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 963 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,5,12,7,13,17,28)(3,15,18,8,16,26,32,27)(4,20,22,9,23,10,19,30)(6,14,25,31,21,11,24,29);(2,8,13,27)(3,16)(4,18)(5,24)(9,28,30,12)(10,14,20,11)(15,31,26,29)(17,25)(19,22)(23,32);(1,3)(2,9)(4,17)(5,23)(6,19)(7,16)(8,12)(10,29)(11,15)(13,30)(14,26)(18,24)(20,31)(21,22)(25,32)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,d^b=dh,f^b=fghi,f^c=fhi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72237360, #72237395, #72237399 |
| maximal normal subgroup of | #72237360, #72237395, #72237399 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40341);