Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,15,22,16,12,14,19)(11,20,24,17,18,23,21,13);(2,6)(4,7)(9,11,16,18)(10,17,12,13)(14,24,15,21)(19,23,22,20);(1,2)(3,4)(5,6)(7,8)(9,12)(10,16)(11,17)(13,18)(14,22)(15,19)(20,21)(23,24);(1,3,5,8)(2,7,6,4)(9,13,16,17)(10,18,12,11)(14,23,15,20)(19,24,22,21);(1,4,5,7)(2,3,6,8)(11,18)(13,17)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524673);