Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1701 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,25)(21,24)(22,26)(27,32)(28,30)(29,31);(1,3,5,4)(2,6,7,8)(17,19,21,29)(18,23,24,31)(20,22,27,28)(25,26,32,30);(9,11,12,16)(10,13,14,15)(18,24)(19,27)(20,29)(22,28)(23,25)(31,32);(1,4,5,3)(2,6,7,8)(17,20,21,27)(18,25,24,32)(19,28,29,22)(23,30,31,26);(19,20)(22,28)(23,25)(26,30)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72282526, #72282894, #72282936, … |
| maximal normal subgroup of | #72282526, #72282894, #72282936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493496);