Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,12,8,10)(17,25)(21,28)(23,26)(24,27);(5,6)(7,10)(8,12)(9,11)(13,14,19,22)(15,16,20,18)(17,26,25,23)(21,24,28,27);(1,2,3,4)(13,15,18,14,19,20,16,22)(17,26,27,28,25,23,24,21);(5,7,9,8)(6,10,11,12)(13,16,19,18)(14,20,22,15)(17,24,25,27)(21,26,28,23);(5,8,9,7)(6,10,11,12)(13,17,16,24,19,25,18,27)(14,21,15,23,22,28,20,26)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724053);