Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,30)(29,32);(5,6,8,7)(10,14)(15,16)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(5,6,8,7)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32);(2,4)(5,7,8,6)(9,12,13,11)(10,16,14,15)(19,27)(23,25)(26,29)(31,32)Polycyclic presentation:
2^9|b^2=fi,c^2=ei,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304199);