Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500010 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,17,19,14)(12,15)(13,16)(18,22,20,23)(21,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24);(2,5)(3,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20);(1,3,4,7)(2,6,5,8)(12,21)(15,24)(18,20)(22,23);(2,5)(6,8)(11,19)(14,17)(18,20)(22,23)Polycyclic presentation:
2^9|a^2=g,d^2=hi,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225735, #72228200, #72233021, … |
| maximal normal subgroup of | #72225735, #72228200, #72233021, … |
| maximal quotient of | #72228194, #72264902 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424035);