Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1970 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)(7,10)(8,9)(11,12)(13,14,17,15,18,21,24,19)(16,20,23,22,25,27,28,26);(1,2)(3,4)(14,19)(15,21)(16,23)(17,24)(22,26)(25,28);(2,4)(6,9)(10,12)(13,15,17,21,18,19,24,14)(16,22,23,27,25,26,28,20);(2,4)(13,16,24,28,18,25,17,23)(14,20,19,26,21,27,15,22);(5,7,8,11)(6,10,9,12)(16,25)(20,27)(22,26)(23,28)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=fi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363852);