Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,23,28,22,27)(19,29,31,32,30,20,25,24);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,26)(21,25)(22,31)(23,30)(27,29)(28,32);(11,15)(13,16)(17,18,23,28)(19,24,30,32)(20,25,29,31)(21,27,22,26);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20)(18,25)(19,26)(21,24)(22,32)(23,29)(27,30)(28,31);(19,30)(20,29)(24,32)(25,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341444);