Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #375718 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14169 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,15,22)(4,16,18,25)(5,9)(6,11)(7,14,23,20)(8,17,24,30)(10,19)(13,27)(21,29)(26,31)(28,32);(1,3,6,15)(2,7)(4,13,18,29)(5,14,19,20)(8,21)(9,12)(10,22)(11,23)(16,26)(17,28,30,31)(24,27)(25,32);(1,2)(3,12,15,22)(4,8)(5,10)(6,11)(7,20,23,14)(9,19)(13,26,29,32)(16,30)(17,25)(18,24)(21,31,27,28);(1,4,5,17,6,18,19,30)(2,8,10,25,11,24,9,16)(3,13,14,28,15,29,20,31)(7,21,22,32,23,27,12,26)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fh,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72215972, #72216507, … |
| maximal normal subgroup of | #72213409, #72215972, #72216507, … |
| maximal quotient of | #72224701, #72224702, #72224703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282914);