Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #205751 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5,7,8)(4,6)(10,14)(11,17)(12,18)(15,22)(16,21)(19,23);(1,3)(2,5,6,8)(4,7)(9,10,12,14,13,16,18,21)(11,15,17,20,19,22,23,24);(2,6)(5,8)(9,11,18,23,13,19,12,17)(10,15,21,24,16,22,14,20);(3,7)(5,8)(10,16)(14,21)(15,22)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fh,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310544, #72310549, #72310552, … |
| maximal normal subgroup of | #72310544, #72310549, #72310552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112948);