Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432953 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62844 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,18,15,12)(13,21)(14,17)(16,22)(19,20,23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23);(2,5)(6,8)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24);(1,3,4,7)(2,6,5,8)(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24);(13,22)(16,21)(19,23)(20,24)Polycyclic presentation:
2^9|a^2=fg,d^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318181 |
| maximal normal subgroup of | #72318181 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342595);