Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142232 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 676 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) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,19,24)(15,25,30,39)(16,31,35,22)(17,36,33,21)(18,37,34,23)(20,26,42,27)(28,41,32,44)(29,40,38,43);(6,9)(7,12)(10,11)(13,15,19,30)(14,20,24,42)(16,32,35,28)(17,26,33,27)(18,38,34,29)(21,43,36,40)(22,39,31,25)(23,44,37,41);(2,4)(5,6)(7,10)(8,11)(9,12)(15,26)(16,33)(17,35)(18,34)(20,39)(21,31)(22,36)(23,37)(25,42)(27,30)(28,38)(29,32)(40,44)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50021);