Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632036 |
| 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 | #24384 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1919 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,28)(21,25)(22,23)(24,27)(29,32)(30,31);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,28,31,32);(11,15)(13,16)(17,20,21,29)(18,24,25,31)(19,27,22,30)(23,28,26,32);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,25)(23,26)(24,31)(28,32);(2,5)(6,7)(10,14)(13,16)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539871);