Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143214 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28233 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6,8,11)(7,9,12,10)(13,14)(15,26)(16,25)(17,21)(18,24)(19,22)(20,27)(23,33)(28,41)(29,40)(30,37)(31,39)(32,36)(34,38)(35,44)(42,43);(1,2)(3,4)(6,9)(7,12)(10,11)(13,15)(14,20)(16,30)(17,32)(18,29)(19,27)(21,35)(22,40)(23,41)(24,37)(25,26)(28,34)(31,43)(33,44)(36,39)(38,42);(2,4)(14,21)(16,19)(17,33)(20,35)(22,39)(23,25)(24,38)(26,41)(27,30)(28,43)(31,34)(32,44)(36,40)(37,42)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,d^c=di,e^c=eh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50961);