Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,19)(18,23,24,22);(1,2)(3,4)(5,7,8,6);(2,4)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=efi,b^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72267126, #72267129, #72267131, … |
| maximal normal subgroup of | #72267126, #72267129, #72267131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302858);