Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907961 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,13)(11,15)(14,16)(17,18)(19,20)(21,24)(22,23);(1,2)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,16,15)(17,19,22,21)(18,20,23,24);(1,3)(2,5)(4,8)(6,7)(9,11,10,14,12,16,13,15)(17,20,19,18,22,24,21,23);(3,8)(5,7)(9,10,12,13)(11,15,16,14)(17,19,22,21)(18,20,23,24);(2,6)(5,7)(11,16)(14,15)(17,19,22,21)(18,20,23,24)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819188);