Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #387441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14473 |
| 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 | |
1190 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)(3,14,19,32)(4,20,22,27)(5,11)(6,12)(7,25)(8,13)(9,17,28,18)(10,16,30,21)(15,26)(23,31)(24,29);(1,3)(2,9)(4,5)(6,18)(7,17)(8,19)(10,29)(11,30)(12,32)(13,28)(14,25)(15,16)(20,31)(21,23)(22,24)(26,27);(1,4,6,21,8,22,7,16)(2,10,12,27,13,30,25,20)(3,15,17,24,19,23,18,5)(9,26,14,29,28,31,32,11);(1,5,7,23,8,24,6,15)(2,11,12,26,13,29,25,31)(3,16,18,22,19,21,17,4)(9,27,14,30,28,20,32,10)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72216142, #72216203, … |
| maximal normal subgroup of | #72213409, #72216142, #72216203, … |
| maximal quotient of | C2×#387441, #72252102, #72314100 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294637);