Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,20)(19,27,31,24)(21,25,23,26)(28,30,32,29);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,31)(18,24,20,27)(21,30,23,29)(25,28,26,32);(10,14)(13,16)(17,20,22,18)(19,28,31,32)(21,25,23,26)(24,29,27,30);(19,29)(21,23)(24,32)(25,26)(27,28)(30,31);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,25,20,26)(19,30,31,29)(24,28,27,32)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349305);