Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,29,32)(20,24,21,25)(27,30,28,31);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,30)(26,32);(1,3,4,7)(2,6,5,8)(13,15)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31);(2,5)(6,8)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=h,c^2=ei,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,f^c=fh,g^c=gh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72309276, #72309277, #72309279, … |
| maximal normal subgroup of | #72309276, #72309277, #72309279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 162330);