Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2396 ◁ 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:
(1,2)(3,4)(5,6,9,11)(7,12,8,10)(13,14)(15,20)(16,21)(17,19)(18,22)(23,26)(24,27)(25,28);(5,6)(7,10)(8,12)(9,11)(14,19)(15,23)(16,20)(18,27)(21,25)(26,28);(2,4)(13,15,17,25)(14,20,19,28)(16,24,26,22)(18,21,27,23);(5,7,9,8)(6,10,11,12)(14,19)(16,26)(20,28)(22,24);(5,8,9,7)(6,10,11,12)(13,16)(14,21)(15,24)(17,26)(18,28)(19,23)(20,27)(22,25)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583447);