Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622880 |
| 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×D4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3758 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)(4,9)(5,10)(6,11)(7,12)(8,15)(14,27)(16,23)(17,22)(18,24)(19,25)(20,26)(21,28)(29,31)(30,32);(1,3,7,4)(2,8,12,22)(5,14,18,16)(6,15,20,17)(9,11,13,26)(10,21,24,31)(19,28,30,29)(23,25,27,32);(1,4,7,3)(2,9,12,13)(5,16,18,14)(6,15,20,17)(8,11,22,26)(10,23,24,27)(19,28,30,29)(21,25,31,32);(1,5)(2,10)(3,14)(4,16)(6,19)(7,18)(8,21)(9,23)(11,25)(12,24)(13,27)(15,28)(17,29)(20,30)(22,31)(26,32);(5,18)(10,24)(14,16)(19,30)(21,31)(23,27)(25,32)(28,29)Polycyclic presentation:
2^9|b^2=i,c^2=i,b^a=bf,c^a=cg,h^a=hi,g^b=ghi,f^c=fh,e^d=ei,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236843, #72237519, #72237528, … |
| maximal normal subgroup of | #72237519, #72237528, #72240644, … |
| maximal quotient of | #72238458, #72255931, #72256627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530078);