Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485605 |
| 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 | #46458 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
905 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 905 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18,22,27)(19,24,20,25)(21,23,30,26)(28,31,29,32);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(18,23)(19,20)(21,30)(24,32)(25,31)(26,27);(17,19)(18,24)(20,22)(21,29)(23,31)(25,27)(26,32)(28,30);(3,10)(6,15)(9,16)(11,13)(17,20,22,19)(18,25,27,24)(21,29,30,28)(23,32,26,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72278427, #72278658, #72318749, … |
| maximal normal subgroup of | #72278427, #72278658, #72318749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392904);