Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14533 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,21,4)(3,12,16,26,15,8,20,13)(5,9,19,25,18,24,31,17)(14,22,29,27,28,23,32,30);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,23)(10,12)(11,26)(16,21)(17,30)(18,28)(19,32)(22,24)(25,27)(29,31);(1,4,6,10,7,11,21,2)(3,13,20,12,15,26,16,8)(5,17,19,24,18,25,31,9)(14,27,32,23,28,30,29,22);(1,5)(2,9)(3,14)(4,17)(6,19)(7,18)(8,22)(10,24)(11,25)(12,23)(13,27)(15,28)(16,29)(20,32)(21,31)(26,30)Polycyclic presentation:
2^9|a^2=h,c^2=fh,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284835, #72284884, #72284899, … |
| maximal normal subgroup of | #72284835, #72284884, #72284899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296187);