Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3082 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)(7,11)(8,19)(9,23)(10,15)(12,16)(13,22)(14,25)(17,28)(18,32)(20,27)(21,35)(24,36)(26,29)(30,33)(31,34);(1,2)(3,4)(6,11)(7,16)(8,20)(12,15)(14,23)(17,22)(18,25)(21,34)(24,33)(27,28)(29,32)(35,36);(2,4)(5,7,10,16)(6,12,15,11)(8,17,22,20)(9,18,26,25)(13,28,19,27)(14,29,32,23)(21,33,24,34)(30,35,31,36);(2,4)(5,8)(6,13)(7,17)(9,24)(10,22)(11,27)(12,28)(14,31)(15,19)(16,20)(18,34)(21,26)(23,35)(25,33)(29,36)(30,32);(5,9)(6,14)(7,18)(8,21)(10,26)(11,23)(12,29)(13,30)(15,32)(16,25)(17,33)(19,31)(20,34)(22,24)(27,36)(28,35)Polycyclic presentation:
2^9|c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,e^d=ei,h^d=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421359);