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 | |
1592 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)(19,23)(20,22)(21,24)(25,28)(26,30)(27,29)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,15,12,14)(19,26)(23,30)(27,32)(29,31);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(2,6)(5,7)(17,19,21,26)(18,23,24,30)(20,27,28,32)(22,29,25,31);(9,12,13,11)(10,15,16,14)(18,24)(19,26)(22,25)(27,32)Polycyclic presentation:
2^9|b^2=fh,c^2=fh,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 | #72257297, #72257512, #72257618 |
| maximal normal subgroup of | #72257297, #72257512, #72257618 |
| maximal quotient of | #72308630 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6759759);