Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437593 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5975 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)(3,14)(4,18)(5,11)(6,24)(7,13)(8,12)(9,15)(10,21)(16,28)(17,29)(19,32)(20,27)(22,31)(23,26)(25,30);(1,3)(2,9)(4,19)(5,20)(6,16)(7,15)(8,17)(10,30)(11,31)(12,28)(13,29)(14,24)(18,26)(21,23)(22,25)(27,32);(1,4)(2,10)(3,5)(6,20)(7,21)(8,22)(9,26)(11,28)(12,27)(13,31)(14,32)(15,19)(16,25)(17,23)(18,24)(29,30);(1,5)(2,11)(3,4)(6,19)(7,25)(8,23)(9,27)(10,28)(12,26)(13,30)(14,18)(15,20)(16,21)(17,22)(24,32)(29,31);(2,12)(9,28)(10,27)(11,26)(13,24)(14,29)(18,31)(30,32)Polycyclic presentation:
2^9|f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215784, #72215958, #72215968, … |
| maximal normal subgroup of | #72215784, #72215958, #72215968, … |
| maximal quotient of | C2×#6437593, #72237517, #72239639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344789);