Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437445 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3250 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,23)(22,24);(10,16)(13,14)(17,24)(18,19)(20,23)(21,22);(2,5)(6,8)(9,11,15,12)(10,17,19,20)(13,21,14,22)(16,23,18,24);(9,12,15,11)(10,17,19,20)(13,22,14,21)(16,23,18,24);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,19,16)(11,21,12,22)(17,24,20,23)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280219, #72281527, #72281573, … |
| maximal normal subgroup of | #72280219, #72281527, #72281573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361041);