Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #38914 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1553 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)(18,22)(20,27)(23,28)(24,29)(26,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16);(3,8)(5,7)(10,13)(11,15)(17,18,20,22,21,24,27,29)(19,23,26,28,25,30,31,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(10,13)(14,16)(18,24)(19,25)(22,29)(26,31)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6847157);