Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,32,28,31);(3,8)(5,7)(10,13)(14,15)(18,23)(19,22)(24,31)(25,26)(27,29)(28,32);(2,6)(5,7)(17,20,21,30)(18,24,25,28)(19,27,22,29)(23,31,26,32);(9,11,12,16)(10,14,13,15)(18,25)(20,30)(23,26)(27,29)Polycyclic presentation:
2^9|b^2=fhi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251266, #72256843, #72256918, … |
| maximal normal subgroup of | #72251266, #72256843, #72256918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6707518);