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 | |
3148 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)(9,10,15,13)(11,19,22,24)(12,18,14,17)(16,23,21,20);(3,7)(5,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,19)(18,24);(1,2)(3,8)(4,6)(5,7)(9,12,15,14)(10,17,13,18)(11,20,22,23)(16,19,21,24);(9,13,15,10)(11,21,22,16)(12,18,14,17)(19,20,24,23);(9,12,15,14)(10,18,13,17)(11,20,22,23)(16,24,21,19)Polycyclic presentation:
2^9|a^2=h,c^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 | #72283125, #72283193, #72283202, … |
| maximal normal subgroup of | #72283125, #72283193, #72283202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260548);