Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131997 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #51902 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
991 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 991 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,15,12,17)(13,18)(14,19)(16,20,23,21)(22,24);(1,2)(3,7)(4,6)(5,8)(10,15)(11,12)(16,24)(17,19)(18,23)(20,21);(3,8)(5,7)(9,11,14,12)(10,16,19,23)(13,21,22,20)(15,18,17,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,e^c=ei,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72213362, #72213687, #72213697, … |
| maximal normal subgroup of | #72213362, #72213687, #72213697, … |
| maximal quotient of | #72338272, #72339621 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39197);