Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| 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 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14,18,22)(12,19)(13,16)(15,20)(17,21,24,23);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14,16,22)(12,17,20,24)(13,18)(15,23)(19,21);(2,6)(5,8)(9,12)(10,15,16,19)(11,17,18,24)(13,20)(14,21)(22,23);(3,7)(5,8)(14,22)(15,19)(17,24)(21,23)Polycyclic presentation:
2^9|a^2=efg,b^2=eh,c^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72225672, #72225679, #72225682, … |
| maximal normal subgroup of | #72225672, #72225679, #72225682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386952);