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×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1594 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)(17,18)(19,28)(20,27)(21,23)(22,26)(24,30)(25,29)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(18,23)(19,29)(26,27)(30,31);(3,8)(5,7)(10,13)(14,15)(17,19)(18,24)(20,30)(21,29)(22,31)(23,32)(25,27)(26,28);(2,6)(5,7)(17,19)(18,25)(20,31)(21,29)(22,30)(23,28)(24,27)(26,32);(9,11,12,16)(10,14,13,15)(17,20,21,22)(18,26,23,27)(19,30,29,31)(24,28,32,25)Polycyclic presentation:
2^9|b^2=fh,e^2=fhi,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 | #72281470 |
| maximal normal subgroup of | #72281470 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6707509);