Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2054 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)(13,14)(15,18)(16,19)(17,20)(21,26)(22,25)(23,27)(24,28);(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,23)(17,21)(19,27)(20,26)(22,24)(25,28);(1,2,3,4)(14,19)(15,21)(17,24)(18,25)(22,23)(26,27);(5,7,9,8)(6,10,11,12);(5,8,9,7)(6,10,11,12)(14,19)(15,22)(17,24)(18,26)(21,23)(25,27)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593926);