Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,28,18,22,17,21)(15,23,24,20,26,27,25,16);(5,6,10,9)(7,12,8,11)(13,15)(14,20)(16,22)(17,24)(18,26)(19,25)(21,27)(23,28);(2,4)(5,7,10,8)(6,11,9,12)(13,16,18,20)(14,15,22,26)(17,27,19,23)(21,25,28,24);(5,8,10,7)(6,11,9,12)(13,17,18,19)(14,21,22,28)(15,24,26,25)(16,23,20,27);(5,9,10,6)(7,12,8,11)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480639);