Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #56343 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
941 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 941 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,25)(19,23,26,32)(20,28,30,24)(22,29,31,27);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,21,26)(18,22,25,31)(20,29,30,27)(23,24,32,28);(3,10)(4,12)(6,13)(9,15)(17,19)(18,23)(20,29)(21,26)(22,28)(24,31)(25,32)(27,30);(3,10)(6,13)(9,15)(11,16)(19,26)(22,31)(23,32)(27,29)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72227670, #72227695 |
| maximal normal subgroup of | #72227670 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394415);