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×Q8 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3681 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)(3,8)(4,9)(5,7)(6,10)(11,21)(12,18)(13,22)(14,23)(15,19)(16,24)(17,20)(25,29)(26,30)(27,31)(28,32);(2,7)(8,18)(9,19)(10,20)(21,29)(22,30)(23,31)(24,32);(5,17)(12,26)(15,28)(25,27);(1,3,6,13)(2,8,10,22)(4,14,16,11)(5,12,17,26)(7,18,20,30)(9,23,24,21)(15,27,28,25)(19,31,32,29);(1,4,6,16)(2,9,10,24)(3,11,13,14)(5,15,17,28)(7,19,20,32)(8,21,22,23)(12,25,26,27)(18,29,30,31)Polycyclic presentation:
2^9|d^2=i,e^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 | #72236916, #72237525, #72237528, … |
| maximal normal subgroup of | #72237525, #72237528, #72302919, … |
| maximal quotient of | #72303228, #72306416, #72306421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530079);