Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232393 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,9,12)(7,8,11,10)(13,14,16,21)(15,23,25,19)(17,20,18,22)(24,28,26,27);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15,16,25)(14,19,21,23)(17,26,18,24)(20,28,22,27);(1,3,4,2)(6,11)(7,12)(8,10)(14,20)(15,24)(17,18)(21,22)(25,26)(27,28);(5,8,9,10)(6,7,12,11)(14,21)(15,25)(20,22)(24,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eh,d^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gh,d^c=di,f^c=fh,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145738);