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 | |
1733 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)(9,10,13,21,15,11,14,18)(12,20,24,16,23,17,22,19);(2,5)(3,7)(9,11)(10,15)(12,16)(13,18)(14,21)(17,24)(19,23)(20,22);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,16,11,19)(13,22,14,24)(17,21,20,18);(2,5)(6,8)(9,12)(10,17)(11,20)(13,24)(14,22)(15,23)(16,18)(19,21);(1,3,4,7)(2,6,5,8)(10,11)(16,19)(17,20)(18,21)Polycyclic presentation:
2^9|a^2=gi,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,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, 7055314);