Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(1,2,3,5,4,6,7,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(2,6)(5,8)(11,18)(14,22)(17,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96024);