Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C25 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,15)(4,10)(5,11)(6,12)(7,13)(8,14)(9,17)(16,28)(18,26)(19,27)(20,29)(21,24)(22,30)(23,31)(25,32);(1,3,8,6)(2,9,14,24)(4,18,5,16)(7,17,22,21)(10,25,11,23)(12,13,15,30)(19,32,20,31)(26,29,28,27);(1,4,8,5)(2,10,14,11)(3,16,6,18)(7,20,22,19)(9,23,24,25)(12,26,15,28)(13,27,30,29)(17,32,21,31);(1,5,8,4)(2,11,14,10)(3,16,6,18)(7,20,22,19)(9,23,24,25)(12,26,15,28)(13,29,30,27)(17,31,21,32);(1,6,8,3)(2,12,14,15)(4,18,5,16)(7,21,22,17)(9,13,24,30)(10,26,11,28)(19,32,20,31)(23,27,25,29)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ci,g^b=ghi,f^c=fh,e^d=ei,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72302917, #72303038, #72303043, … |
| maximal normal subgroup of | #72302917, #72303038, #72303043, … |
| maximal quotient of | #72312597, #72312610, #72312827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530087);