Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7211759 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #45599 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,9,6,10,20,18)(3,7,13,22,14,11,19,24)(4,8,16,15,17,23,29,30)(12,21,27,32,28,25,31,26);(1,2)(3,11)(4,15)(5,18)(6,10)(7,14)(8,16)(9,20)(12,26)(13,22)(17,30)(19,24)(21,31)(23,29)(25,27)(28,32);(1,3,5,19,6,14,20,13)(2,7,18,22,10,11,9,24)(4,12,16,31,17,28,29,27)(8,21,30,32,23,25,15,26);(1,4,5,16,6,17,20,29)(2,8,18,30,10,23,9,15)(3,12,19,31,14,28,13,27)(7,21,22,32,11,25,24,26);(1,2)(3,7)(4,8)(5,18)(6,10)(9,20)(11,14)(12,21)(13,24)(15,29)(16,30)(17,23)(19,22)(25,28)(26,27)(31,32)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi,f^e=fi,g^e=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7118955);