Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10,14,18,15,12,13,17)(11,19,22,16,23,24,21,20);(1,2)(3,4)(5,6,8,7)(9,11,15,23)(10,16,12,20)(13,22,14,21)(17,24,18,19);(2,4)(9,10)(11,20)(12,15)(13,18)(14,17)(16,23)(19,21)(22,24);(5,7,8,6)(9,12)(10,15)(11,20)(13,18)(14,17)(16,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=efi,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308247);