Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,25)(21,26)(22,31)(23,29)(24,30)(28,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18,20,25)(19,28,29,24)(21,31,22,26)(23,32,27,30);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,19)(18,23)(20,29)(21,32)(22,30)(24,26)(25,27)(28,31);(3,8)(6,7)(11,16)(13,15)(17,19)(18,24)(20,29)(21,30)(22,32)(23,26)(25,28)(27,31)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332277);