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 | #42982 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1450 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(17,18)(19,26)(20,24)(21,23)(22,28)(25,30)(27,32)(29,31);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,29)(21,27)(23,31)(24,26)(25,32)(28,30);(10,13)(14,16)(18,23)(19,27)(24,25)(28,29);(1,4,5,3)(2,8,7,6)(18,24)(20,30)(22,26)(23,25)(28,29)(31,32)Polycyclic presentation:
2^9|b^2=g,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6623696);