Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| 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 | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2003 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)(19,27)(20,25)(21,26)(22,24)(23,28)(29,31)(30,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,29)(21,28)(22,30)(24,31)(25,32)(26,27);(10,14)(13,16)(19,28)(23,27)(29,30)(31,32);(18,24)(20,22)(23,31)(25,26)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,25,26,24)(19,29,28,30)(23,32,27,31)Polycyclic presentation:
2^9|b^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,c^b=ci,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72251269, #72256846, #72256926, … |
| maximal normal subgroup of | #72251269, #72256846, #72256926, … |
| maximal quotient of | #72308459 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537087);