Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2988 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)(3,11)(4,10)(5,8)(6,9)(7,12)(13,16)(14,15)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(20,29)(24,28)(25,32)(27,30);(17,19)(18,22)(20,30)(21,26)(23,31)(24,32)(25,28)(27,29);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,11,15,16)(18,23)(22,31)(24,28)(25,32);(2,9)(3,13)(8,10)(12,14)Polycyclic presentation:
2^9|d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72306994, #72307001, #72307008, … |
| maximal normal subgroup of | #72306994, #72307001, #72307008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256727);