Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
521 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,20,16)(15,25,30,26)(17,23,36,33)(18,22,37,32)(19,34,35,24)(21,39,31,27)(28,42,44,41)(29,43,38,40);(2,4)(6,10)(7,8)(11,12)(13,15,20,30)(14,21,16,31)(17,28,36,44)(18,27,37,39)(19,38,35,29)(22,40,32,43)(23,25,33,26)(24,42,34,41);(6,11)(7,8)(10,12)(13,16,20,14)(15,26,30,25)(17,33,36,23)(18,32,37,22)(19,24,35,34)(21,27,31,39)(28,41,44,42)(29,40,38,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fghi,g^b=gi,d^c=dg,e^c=eh,f^c=fghi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51324);