Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130062 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #2815 | C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,13,18,14,17,29,34)(11,19,22,36,20,16,27,33)(12,23,26,35,25,15,28,31)(21,30,37,39,24,32,38,40);(1,2)(3,7)(4,6)(5,8)(10,15)(11,20)(16,32)(17,23)(18,35)(19,30)(21,24)(22,27)(31,34)(33,39)(36,40)(37,38);(9,11,14,20)(10,16,17,19)(12,24,25,21)(13,27,29,22)(15,30,23,32)(18,36,34,33)(26,37,28,38)(31,40,35,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fi,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72318212, #72318458, #72318719, … |
| maximal normal subgroup of | #72318212, #72318458, #72318719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37268);