Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 473 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) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14,18,24,33,42,17,21,20,26,16,23,19,25,34,38)(15,27,30,22,36,37,35,44,32,43,28,41,31,39,29,40);(1,2,3,4)(14,21)(16,17)(18,34)(19,33)(22,37)(23,25)(24,42)(26,38)(27,44)(28,35)(29,30)(31,36)(39,41)(40,43);(6,9)(7,12)(10,11)(13,15)(14,22)(16,29)(17,30)(18,35)(19,36)(20,32)(21,37)(23,27)(24,43)(25,44)(26,41)(28,34)(31,33)(38,39)(40,42)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=gh,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44738);