Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,23,21,24,19,20);(5,7,8,6)(9,11,12,16)(10,13,14,15)(18,23)(19,22)(20,24);(6,7)(10,14)(13,15)(17,19,21,22)(18,20,24,23);(2,4)(6,7)(11,16)(13,15)(17,20,22,18,21,23,19,24)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=i,d^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264567, #72264584, #72264597, … |
| maximal normal subgroup of | #72264567, #72264584, #72264597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343856);