Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7,12,14)(5,8)(10,13)(15,16)(18,23,25,28)(24,31,32,30);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,21,23,22,25,20,28)(19,24,26,30,27,32,29,31);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(17,19)(18,24)(20,29)(21,26)(22,27)(23,30)(25,32)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239053 |
| maximal normal subgroup of | #72239053 |
| maximal quotient of | #72239114, #72239116 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17918);