Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,14,11)(12,16,19,18)(13,15,21,17)(20,22,24,23);(5,6)(7,8)(9,11,14,10)(12,18,19,16)(13,17,21,15)(20,23,24,22);(2,4)(5,7,8,6)(10,15)(11,17)(13,21)(16,22)(18,23)(20,24);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24);(12,19)(16,18)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339108);