Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4032 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)(3,6)(4,5)(7,8)(9,10,14,12)(11,17,20,24)(13,21,23,16)(15,22,18,19);(2,5)(6,8);(9,11)(10,15)(12,18)(13,22)(14,20)(16,24)(17,21)(19,23);(9,12,14,10)(11,18,20,15)(13,16,23,21)(17,22,24,19);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,h^c=hi,g^d=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72312618, #72312669 |
| maximal normal subgroup of | #72312618, #72312669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536558);