Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470067 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(10,15)(12,13)(16,22)(17,21)(19,20)(23,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,21)(13,15)(14,17)(18,23)(19,24)(20,22);(2,5)(6,8)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,22)(20,24);(2,5)(6,8)(9,11,14,18)(10,16,17,23)(12,19,13,20)(15,22,21,24);(1,3,4,7)(2,6,5,8)(11,18)(16,23)(19,20)(22,24)Polycyclic presentation:
2^9|d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398174);