Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #42532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(18,22)(19,25)(20,28)(24,29)(27,30)(31,32);(10,14)(13,15)(17,18,21,24)(19,26,27,23)(20,29,28,22)(25,32,30,31);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(19,27)(23,26)(25,30)(31,32);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(17,19)(18,23)(20,30)(21,27)(22,31)(24,26)(25,28)(29,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6649028);