Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,11)(10,14)(12,17)(15,23)(18,31)(19,28)(20,30)(21,25)(22,29)(24,27)(26,32)(33,35)(34,36);(2,4)(6,11)(7,17)(8,20)(9,23)(14,15)(16,28)(19,24)(21,22)(25,30)(26,31)(32,33)(35,36);(5,7,10,19)(6,12,14,27)(8,18,22,34)(9,24,23,17)(11,16,15,28)(13,26,29,35)(20,32,21,33)(25,31,30,36);(5,8)(6,13)(7,18)(9,21)(10,22)(11,25)(12,26)(14,29)(15,30)(16,31)(17,32)(19,34)(20,23)(24,33)(27,35)(28,36);(6,14)(11,15)(12,27)(13,29)(16,28)(25,30)(26,35)(31,36)Polycyclic presentation:
2^9|c^2=i,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,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237505, #72256282, #72256289, … |
| maximal normal subgroup of | #72237505, #72256282, #72256289, … |
| maximal quotient of | #72310302, #72310303, #72310308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344800);