Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417692 |
| 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 | |
1101 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)(11,13)(12,14)(15,16)(17,18,23,26,22,25,32,27)(19,21,24,30,28,20,29,31);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,25)(22,28)(23,24)(26,31)(27,30)(29,32);(10,14)(13,16)(17,20)(18,24)(19,27)(21,22)(23,31)(25,29)(26,28)(30,32);(1,4,5,2)(3,6,8,7)(17,21)(18,19)(20,22)(23,31)(24,27)(25,28)(26,29)(30,32)Polycyclic presentation:
2^9|a^2=ehi,b^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 357315);