Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1813 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,23,28,32)(20,25,21,30)(24,29,31,27);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,3,5,4)(2,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31);(1,4,5,3)(2,6,8,7)(11,15)(13,16)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=f,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72257444, #72257531, #72264360, … |
| maximal normal subgroup of | #72257444, #72257531, #72264360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6501254);