Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574100 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61402 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,17)(11,13,16,23)(12,19,22,18)(15,20,24,21);(2,6)(4,7)(9,11,13,17,14,16,23,10)(12,18,20,24,22,19,21,15);(1,2)(3,4)(5,6)(7,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,24)(21,23);(1,3,5,8)(2,7,6,4)(9,13,14,23)(10,16,17,11)(12,20,22,21)(15,19,24,18);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6525016);