Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480250 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1234 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)(11,18)(12,15)(13,17)(14,22,21,16)(19,24,20,23);(2,5)(6,8)(9,11,13,12)(10,15,17,18)(14,19,21,20)(16,24,22,23);(10,16)(11,12)(15,23)(17,22)(18,24)(19,20);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,17,18)(14,20,21,19)(16,24,22,23)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=e,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^c=di,g^c=gh,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312002, #72312007, #72312038, … |
| maximal normal subgroup of | #72312002, #72312007, #72312038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401205);