Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1133 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)(9,10,14,11)(12,19,20,23)(13,15,21,17)(16,24,18,22);(2,4)(5,7,8,6)(10,15,11,17)(12,20)(16,19,18,23)(22,24);(2,4)(9,11)(10,14)(12,18)(13,15)(16,20)(17,21)(19,24)(22,23);(2,4)(9,12,21,24,14,20,13,22)(10,16,15,23,11,18,17,19)Polycyclic presentation:
2^9|a^2=h,b^2=egh,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275127);