Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24440 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,23)(21,24)(22,28)(26,29)(27,32)(30,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,20,28)(18,22,23,25)(21,27,30,29)(24,26,31,32);(10,14)(13,16)(17,18)(19,26)(20,23)(21,31)(22,27)(24,30)(25,29)(28,32);(19,27)(21,30)(22,26)(24,31)(25,32)(28,29);(1,3,4,7)(2,6,5,8)(18,23)(22,25)(24,31)(26,32)Polycyclic presentation:
2^9|b^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239656, #72256835, #72256872, … |
| maximal normal subgroup of | #72239656, #72256835, #72256872, … |
| maximal quotient of | #72239634 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349382);