Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636816 |
| 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 | #2856 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,15)(11,13,16,12)(17,18,21,24)(19,25,27,22)(20,28,30,23)(26,31,29,32);(1,3)(2,5)(4,7)(6,8)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31);(3,7)(5,8)(9,11,13,15,14,16,12,10);(3,7)(5,8)(9,12,14,13)(10,11,15,16)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32);(2,6)(5,8)(18,24)(22,25)(23,28)(31,32)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6545425);