Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8,6,9,4,13)(3,10,11,15,12,7,14,16)(17,18)(19,27)(20,26)(21,25)(22,24)(23,28)(29,32)(30,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(18,24)(23,31)(25,26)(27,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32);(17,20,22,21)(18,26,24,25)(19,28,29,30)(23,32,31,27)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=i,g^2=hi,b^a=bf,c^a=ci,d^a=dh,f^a=fh,f^b=fi,g^b=ghi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235050, #72240112, #72306719, … |
| maximal normal subgroup of | #72235050, #72240112, #72306719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251252);