Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7316 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
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 | |
1376 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1376 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:
(1,2,5,9)(3,11,13,23)(4,8,15,21)(6,10,18,24)(7,17,20,14)(12,25,26,31)(16,22,29,32)(19,28,30,27);(1,3)(2,7,10,11)(4,12)(5,17,18,13)(6,14)(8,19,22,25)(9,23)(15,28,29,26)(16,27)(20,24)(21,31)(30,32);(1,4,6,16)(2,8,10,22)(3,12,14,27)(5,15,18,29)(7,19,11,25)(9,21,24,32)(13,26,17,28)(20,30,23,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,b^a=bd,d^a=dg,f^a=fh,h^a=hi,d^b=dh,e^b=eg,g^b=gi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233458, #72233461, #72240478, … |
| maximal normal subgroup of | #72233458, #72233461, #72240478, … |
| maximal quotient of | C4×#7316, C2×#128024 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35220);