Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14019 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,11,6,4)(3,13,17,25,15,9,16,18)(5,20,22,26,21,10,23,19)(14,24,28,32,29,27,30,31);(1,3,7,16,8,15,6,17)(2,9,12,25,11,13,4,18)(5,14,23,28,21,29,22,30)(10,24,26,32,20,27,19,31);(1,4,6,11,8,12,7,2)(3,13,16,18,15,9,17,25)(5,19,22,20,21,26,23,10)(14,27,28,31,29,24,30,32);(1,5)(2,10)(3,14)(4,19)(6,22)(7,23)(8,21)(9,24)(11,20)(12,26)(13,27)(15,29)(16,28)(17,30)(18,31)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320057, #72338478, #72338488, … |
| maximal normal subgroup of | #72320057, #72338478, #72338488, … |
| maximal quotient of | #72343052 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275663);