Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,16)(8,13)(9,15)(10,22)(11,14)(12,18)(17,32)(19,27)(20,29)(21,28)(23,26)(24,30)(25,34)(31,35)(33,36);(1,2)(3,4)(5,7)(6,12)(8,17)(9,18)(10,23)(11,19)(13,25)(14,26)(15,27)(16,22)(20,33)(21,35)(24,34)(28,36)(29,32)(30,31);(2,4)(5,8)(6,13)(7,17)(9,20)(10,24)(11,21)(12,25)(14,28)(15,29)(16,31)(18,33)(19,35)(22,30)(23,34)(26,36)(27,32);(1,2)(3,4)(6,14)(12,26)(13,28)(15,22)(16,27)(25,36)(29,30)(31,32);(2,4)(5,9,11,10)(6,15,14,22)(7,18,19,23)(8,20,21,24)(12,27,26,16)(13,29,28,30)(17,33,35,34)(25,32,36,31)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480871);