Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518936 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,6,8,7)(9,10,14,11)(12,16,13,15)(17,18,22,19)(20,21,24,23);(1,3)(2,6)(4,7)(5,8)(9,11,14,10)(12,15,13,16)(17,19,21,24,22,18,23,20);(1,4,5,2)(3,7,8,6)(9,12,14,13)(10,15,11,16);(3,8)(6,7)(9,13,14,12)(10,16,11,15);(17,20,21,19,22,24,23,18)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=h,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7513002);