Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 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)(9,10)(11,20)(12,22)(13,18)(14,24)(15,19)(16,17)(21,23);(1,3,4,2)(9,11,13,17)(10,16,18,20)(12,19,23,24)(14,21,15,22);(5,7,8,6)(9,12,13,23)(10,17,18,11)(14,20,15,16)(19,21,24,22);(2,3)(9,11)(10,12)(13,17)(14,22)(15,21)(16,24)(18,23)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295437);