Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C25 | D8 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 774 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,18,23,33,42,31,40,19,24,17,22,16,21,32,41)(15,25,28,38,26,44,34,20,30,37,27,36,29,43,35,39);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,26)(17,34)(18,35)(19,30)(21,36)(22,43)(23,44)(24,39)(25,40)(27,31)(28,32)(29,33)(37,41)(38,42);(5,7)(6,9)(8,12)(10,11)(13,16,19,33)(14,21,24,42)(15,26,30,29)(17,31,18,32)(20,36,39,38)(22,40,23,41)(25,44,37,43)(27,35,28,34)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44341);