Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #273051 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1093 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)(5,6,10,7)(8,9,11,12)(13,14,17,20)(15,21,23,18)(16,19,25,27)(22,26,24,28);(1,2,4,3)(5,7,10,6)(8,12,11,9)(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,26,27,28);(5,8,9,7,10,11,12,6)(13,16)(14,19)(15,22,23,24)(17,25)(18,26,21,28)(20,27);(1,3,4,2)(14,20)(16,25)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=f,d^2=ehi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dhi,g^c=ghi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 204741);