Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144228 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,30,29,32);(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,27)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,e^b=eh,e^c=ei,h^c=hi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72316687, #72316689, #72316699, … |
| maximal normal subgroup of | #72316687, #72316689, #72316699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51483);