Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(6,10)(7,8)(11,12)(13,14)(15,20)(16,26)(17,21)(18,23)(19,24)(22,27)(25,28);(1,2,4,3)(6,11)(7,8)(10,12)(13,15,17,24)(14,19,21,20)(16,18,25,27)(22,28,23,26);(5,6,8,11,9,12,7,10)(13,16,27,24,17,25,18,15)(14,20,23,28,21,19,22,26);(1,3,4,2)(14,21)(19,20)(22,23)(26,28)Polycyclic presentation:
2^9|b^2=e,c^2=fh,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,g^c=ghi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 204599);