Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15268 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,13,5,12)(3,14,21,10,17,23,16,8)(4,9,20,25,19,26,18,27)(15,28,31,22,30,32,29,24);(1,3,5,16,6,17,7,21)(2,8,11,14,13,10,12,23)(4,15,18,29,19,30,20,31)(9,24,25,28,26,22,27,32);(1,4)(2,9)(3,15)(5,20)(6,19)(7,18)(8,22)(10,24)(11,27)(12,25)(13,26)(14,28)(16,31)(17,30)(21,29)(23,32);(1,3)(2,10)(4,15)(5,21)(6,17)(7,16)(8,13)(9,24)(11,14)(12,23)(18,31)(19,30)(20,29)(22,26)(25,32)(27,28)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326388);