Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470092 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,28,32)(20,24,21,25)(23,30,31,29);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(19,28)(23,31)(27,32)(29,30);(11,15)(13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,32)(25,27)(26,31);(11,15)(13,16)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,32,31,27);(10,14)(13,16)(17,20,22,21)(18,25,26,24)(19,29,28,30)(23,27,31,32)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377660);