Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487635 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #18192 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,27)(20,24)(21,30)(22,25)(23,28)(26,31)(29,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,23)(20,22)(21,29)(24,26)(25,32)(27,30)(28,31);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20,22,19)(18,24,25,27)(21,28,31,29)(23,30,32,26);(2,7)(6,15)(8,12)(9,16)(18,25)(19,20)(26,30)(28,29)Polycyclic presentation:
2^9|b^2=gi,c^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72283025 |
| maximal normal subgroup of | #72283025 |
| maximal quotient of | #72284464 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394997);