Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347252 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(10,15)(12,13)(16,23)(17,20)(18,21)(22,24);(2,6)(3,4)(7,8)(9,10)(11,16)(12,15)(13,20)(14,17)(18,22)(19,24)(21,23);(9,11)(10,16)(12,18)(13,21)(14,19)(15,22)(17,24)(20,23);(9,12,14,13)(10,15,17,20)(11,18,19,21)(16,22,24,23);(1,3,5,4)(2,7,8,6)(10,17)(15,20)(16,24)(22,23)Polycyclic presentation:
2^9|d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72263904, #72281528, #72281576, … |
| maximal normal subgroup of | #72263904, #72281528, #72281576, … |
| maximal quotient of | #72264184 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254493);