Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,11)(12,20)(13,14)(17,18)(19,23)(21,22);(5,7,8,6)(9,10)(11,14)(12,16)(13,17)(15,18)(19,20)(21,23)(22,24);(2,4)(9,11,14,10,15,17,13,18)(12,19,20,16,22,23,21,24);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275342);