Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | #38049 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,24)(20,28)(21,25)(22,27)(23,26)(29,30)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,26,30)(20,25,31,27)(21,32,22,28);(3,8)(5,7)(10,13)(14,16)(17,20,21,30,23,31,22,24)(18,19,27,32,26,29,25,28);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,26,27)(19,28,29,32)(20,30,31,24);(10,13)(14,16)(17,21,23,22)(18,25,26,27)(19,28,29,32)(20,30,31,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6692403);