Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7211754 |
| 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 |
| prime radical | #30496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,9,6,10,17,20)(3,11,13,26,14,7,21,23)(4,15,18,16,19,8,24,25)(12,22,29,31,30,27,32,28);(1,2)(3,7)(4,16)(5,20)(6,10)(8,24)(9,17)(11,14)(12,28)(13,26)(15,18)(19,25)(21,23)(22,32)(27,29)(30,31);(1,3,5,21,6,14,17,13)(2,7,20,23,10,11,9,26)(4,12,18,32,19,30,24,29)(8,22,16,31,15,27,25,28);(1,4,17,24,6,19,5,18)(2,8,9,25,10,15,20,16)(3,12,13,29,14,30,21,32)(7,22,26,28,11,27,23,31);(1,2)(3,7)(4,8)(5,20)(6,10)(9,17)(11,14)(12,22)(13,26)(15,19)(16,18)(21,23)(24,25)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,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=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7118953);