Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(14,19)(15,23)(16,26)(18,25)(20,21)(24,27);(1,2)(3,4)(6,9)(7,12)(13,14)(15,24)(16,27)(17,19)(18,22)(20,26)(21,23)(25,28);(2,4)(13,15,17,26)(14,20,19,24)(16,25,23,18)(21,28,27,22);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,25)(17,23)(18,26)(19,27)(20,28)(22,24);(14,19)(20,24)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409502);