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 |
| prime radical | #36062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1894 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)(7,11)(8,12)(9,10)(13,14,19,20)(15,21,17,23)(16,25,27,22)(18,24,26,28);(6,10)(11,12)(13,15)(14,21)(16,26)(17,19)(18,27)(20,23)(22,24)(25,28);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,19,27)(14,22,20,25)(15,24,17,28)(18,23,26,21);(5,7,9,8)(6,11,10,12)(13,17,19,15)(14,23,20,21)(16,18,27,26)(22,28,25,24);(5,8,9,7)(6,12,10,11)(13,18)(14,24)(15,25)(16,23)(17,22)(19,26)(20,28)(21,27)Polycyclic presentation:
2^9|a^2=i,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, 6492772);