Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2121 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)(5,6,9,10)(7,11,8,12)(14,18)(19,20)(21,26)(22,28);(6,10)(11,12)(13,14)(15,22)(16,18)(17,26)(19,23)(20,27)(21,25)(24,28);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15,16,24)(14,19,18,20)(17,27,25,23)(21,28,26,22);(5,7,9,8)(6,11,10,12)(14,18)(15,23)(17,25)(19,28)(20,22)(24,27);(5,8,9,7)(6,12,10,11)(13,15,16,24)(14,20,18,19)(17,27,25,23)(21,22,26,28)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492939);