Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 52 points:
(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14)(21,22,26,31)(23,32,35,50)(24,27,39,43)(25,40,38,30)(28,44,46,33)(29,47,42,36)(34,51,41,48)(37,49,52,45);(1,2)(3,4)(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)(22,27)(24,36)(25,38)(28,32)(29,30)(31,43)(33,37)(34,41)(39,47)(40,42)(44,52)(45,51)(46,50)(48,49);(2,4)(21,23,26,35)(22,28,31,46)(24,37,39,52)(25,41,38,34)(27,32,43,50)(29,48,42,51)(30,49,40,45)(33,47,44,36)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50619);