Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #34981 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,7,10,8,11,12,9)(13,14)(15,21,23,18)(16,24)(17,20)(19,25)(22,27,28,26);(1,2)(3,4)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(2,4)(6,9)(7,12)(10,11)(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28);(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fh,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174856);