Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C25 |
| socle | C2 | C22×B(4,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3718 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)(3,13)(4,10)(5,18)(6,9)(7,12)(8,15)(11,22)(14,25)(16,29)(17,27)(19,24)(20,28)(21,30)(23,31)(26,32);(1,3)(2,8)(4,15)(5,19)(6,21)(7,14)(9,23)(10,25)(11,26)(12,24)(13,18)(16,22)(17,32)(20,31)(27,30)(28,29);(2,9)(6,22)(8,23)(10,27)(11,12)(13,29)(16,21)(17,20)(18,28)(24,26)(25,30)(31,32);(3,14)(8,24)(13,25)(15,19)(16,21)(23,26)(29,30)(31,32);(1,4,7,5)(2,10,12,18)(3,15,14,19)(6,17,22,20)(8,25,24,13)(9,27,11,28)(16,31,21,32)(23,30,26,29)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,d^b=di,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237532, #72237540, #72237552, … |
| maximal normal subgroup of | #72237532, #72237540, #72237552, … |
| maximal quotient of | #72260444, #72261190, #72261231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530099);