Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391879 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14662 |
| 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 | |
1033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,20,8)(3,12,19,23,14,7,15,21)(4,9,17,26,18,27,32,24)(13,28,16,31,29,22,30,25);(1,3)(2,7,11,12)(4,16,18,30)(5,19)(6,14)(8,23,10,21)(9,25)(13,17,29,32)(15,20)(22,24)(26,28)(27,31);(2,8)(5,20)(7,21)(9,24)(10,11)(12,23)(13,29)(15,19)(17,32)(22,31)(25,28)(26,27);(1,4)(2,9)(3,13,14,29)(5,17)(6,18)(7,22,12,28)(8,24)(10,26)(11,27)(15,30,19,16)(20,32)(21,25,23,31)Polycyclic presentation:
2^9|a^2=ef,b^2=h,d^2=e,f^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,f^c=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72216037, #72216516, … |
| maximal normal subgroup of | #72213409, #72216037, #72216516, … |
| maximal quotient of | C2×#391879, #72250462, #72255612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299075);