Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(10,15)(11,18)(12,13)(16,19)(17,23)(20,24);(2,5)(3,7)(9,10,14,17)(11,19,18,24)(12,23,13,15)(16,21,20,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,15)(13,23)(14,17)(16,18)(19,21)(22,24);(2,5)(6,8)(9,11)(10,16)(12,21)(13,22)(14,18)(15,24)(17,20)(19,23);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,15,17,23)(11,21,18,22)(16,24,20,19)Polycyclic presentation:
2^9|b^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050013);