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 | |
3741 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)(3,9)(4,10)(5,11)(6,12)(7,13)(8,15)(14,24)(16,25)(17,26)(18,27)(19,23)(20,21)(22,28)(29,32)(30,31);(1,3,7,5)(2,8,13,23)(4,14,17,18)(6,15,20,19)(9,21,11,12)(10,22,26,32)(16,28,30,29)(24,31,27,25);(1,3,7,5)(2,9,13,11)(4,14,17,18)(6,19,20,15)(8,21,23,12)(10,24,26,27)(16,29,30,28)(22,31,32,25);(1,4,7,17)(2,10,13,26)(3,14,5,18)(6,16,20,30)(8,22,23,32)(9,24,11,27)(12,25,21,31)(15,28,19,29);(1,5,7,3)(2,11,13,9)(4,14,17,18)(6,19,20,15)(8,12,23,21)(10,24,26,27)(16,28,30,29)(22,31,32,25)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,g^b=ghi,f^c=fh,e^d=ei,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236804, #72236997, #72237528, … |
| maximal normal subgroup of | #72236997, #72237528, #72303038, … |
| maximal quotient of | #72303228, #72306416, #72306421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530081);