Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396094 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,14,18,21)(12,19)(13,16)(15,20)(17,23,24,22);(1,3,5,4)(2,6,7,8)(9,11)(10,14,16,21)(12,17)(13,18)(15,22,19,23)(20,24);(14,21)(22,23);(1,4,5,3)(2,6,7,8)(9,12)(10,15,16,19)(11,17)(13,20)(14,22,21,23)(18,24)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225693, #72225695, #72225700, … |
| maximal normal subgroup of | #72225693, #72225695, #72225700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310228);