Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115437 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,5)(4,7)(6,8)(10,13,14,16);(2,5)(3,8)(6,7)(11,15)(13,16);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226228, #72226231, #72226265 |
| maximal normal subgroup of | #72226228, #72226231, #72226265 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22633);