Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403352 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2053 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(1,3,4,7)(2,5,6,8)(9,11)(10,14,16,17)(12,18)(13,19)(15,23,20,22)(21,24);(9,12)(10,15,16,20)(11,18)(13,21)(14,22,17,23)(19,24);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=gh,c^2=g,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225653, #72225654, #72225656, … |
| maximal normal subgroup of | #72225653, #72225654, #72225656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318047);