Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115302 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#2430 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1971 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1971 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2)(3,7,8,5)(4,6)(9,10,13,17,14,18,12,22)(11,19,20,15,21,16,23,24);(5,7)(9,11)(10,15)(12,23)(13,20)(14,21)(16,17)(18,24)(19,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,23)(14,21)(15,17)(18,19)(22,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=dh,d^c=di,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72260374, #72260416, #72260436, … |
| maximal normal subgroup of | #72260374, #72260416, #72260436, … |
| maximal quotient of | C2×#115310 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22506);