Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #25575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,25,27,32)(22,29,30,24)(23,31,26,28);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,21)(22,24)(23,27)(25,31)(28,32)(29,30);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,28,21,27,30,31)(18,23,24,25,20,26,29,32);(3,8)(6,7)(11,16)(13,15)(17,20,21,18)(19,23,27,26)(22,29,30,24)(25,31,32,28);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,20)(23,26)(24,29)(25,32)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6934770);