Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| 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 | C2×#2018 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
773 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 773 conjugacy classes of subgroups — too many to draw. Only its 94 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)(13,14,16,22,17,23,32,37,19,25,33,36,18,24,30,20)(15,26,27,43,34,21,38,39,29,42,31,35,28,40,44,41);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,20)(16,30)(17,18)(21,35)(22,24)(23,36)(25,37)(26,41)(27,44)(28,34)(31,38)(32,33)(39,42)(40,43);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,31)(17,34)(18,28)(19,29)(20,35)(22,41)(23,42)(24,26)(25,40)(27,33)(30,38)(32,44)(36,39)(37,43)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44343);