Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,15)(8,19)(9,11)(10,23)(12,16)(13,21)(14,22)(17,29)(18,28)(20,30)(24,26)(25,27)(31,33)(32,36)(34,35);(1,2,4,3)(6,11)(7,16)(10,22)(12,28)(15,26)(17,33)(18,24)(19,30)(21,25)(29,36)(31,35)(32,34);(5,7)(6,12)(8,17)(9,18)(10,24)(11,26)(13,29)(14,28)(15,23)(16,22)(19,31)(20,34)(21,32)(25,33)(27,35)(30,36);(5,8,9,20)(6,13,11,27)(7,17,18,34)(10,25,22,21)(12,29,26,35)(14,30,23,19)(15,31,28,36)(16,32,24,33);(1,3,4,2)(6,11)(12,26)(13,27)(14,23)(15,28)(19,30)(29,35)(31,36)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425985);