Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16,20,15)(13,17)(18,22,24,21)(19,23);(2,5)(6,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22);(10,15)(12,20)(14,21)(16,17)(18,24)(22,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^c=fi,g^c=ghi,g^f=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72236637, #72236649, #72236706, … |
| maximal normal subgroup of | #72236637, #72236649, #72236706, … |
| maximal quotient of | C4×#12797, C2×#493957 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401153);