Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×H(5,𝔽2) | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,18)(12,21)(13,22)(14,17)(15,20)(16,24)(19,23);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,22)(13,19)(14,20)(16,23)(17,18)(21,24);(9,12)(10,11)(13,15)(14,23)(16,19)(17,20)(18,24)(21,22);(9,11)(10,12)(13,19)(14,20)(15,16)(17,23)(18,22)(21,24);(1,4,5,3)(2,7,8,6)(9,13,14,24)(10,16,17,22)(11,19,20,21)(12,15,23,18)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426499);