Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7310 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1304 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| socle | C2 | C2×#1304 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1396 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,7,12,19)(4,8,14,22)(6,9,17,23)(11,18,24,29)(13,20,16,28)(15,21,27,31)(25,30,26,32);(1,3)(2,7,9,20)(4,11)(5,16,17,12)(6,13)(8,18,21,30)(10,19)(14,26,27,24)(15,25)(22,29)(23,28)(31,32);(1,4,6,15)(2,8,9,21)(3,11,13,25)(5,14,17,27)(7,18,20,30)(10,22,23,31)(12,24,16,26)(19,29,28,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gi,h^a=hi,d^b=dh,e^b=eg,g^b=gi,e^d=ei,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233452, #72233453, #72233461, … |
| maximal normal subgroup of | #72233452, #72233453, #72233461, … |
| maximal quotient of | C4×#7310, C2×#128016 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35212);