Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124874 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1560 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8344 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 625 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,5,10,4,8,17,24,6,11,20,28,18,9,25,26)(3,12,14,29,15,23,30,32,16,21,31,27,13,7,19,22);(1,3)(2,7)(4,13)(5,19)(6,16)(8,21)(9,12)(10,27)(11,23)(14,25)(15,18)(17,31)(20,30)(22,26)(24,32)(28,29);(2,8,11,9)(7,21,23,12)(10,24,28,26)(22,27,32,29)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265701, #72265725 |
| maximal normal subgroup of | #72265701, #72265725 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37238);