Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267755 |
| 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 | C2×#1595 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1452 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)(5,6,8,10)(7,12,11,9)(13,14)(15,21)(16,19)(17,20)(18,23)(22,26)(24,28)(25,27);(1,2,4,3)(6,10)(7,11)(13,15)(14,18)(16,24)(17,23)(19,26)(20,21)(22,25)(27,28);(5,7,9,6,8,11,12,10)(13,16)(14,19)(15,22,23,24)(17,25)(18,26,21,28)(20,27);(1,3,4,2)(6,10)(7,11)(14,20)(18,21)(19,27)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=f,d^2=ehi,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 204506);