Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353301 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,15,19)(11,20,12,17)(13,18,14,23)(16,22,24,21);(3,7)(5,8)(9,11,15,12)(10,16,19,24)(13,21,14,22)(17,23,20,18);(1,2)(3,8)(4,6)(5,7);(9,12,15,11)(10,17,19,20)(13,21,14,22)(16,23,24,18);(9,13,15,14)(10,18,19,23)(11,21,12,22)(16,17,24,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72338416, #72338417, #72338418, … |
| maximal normal subgroup of | #72338416, #72338417, #72338418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260557);