Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 | |
2482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,13,19)(7,14,21,8)(9,24,15,16)(10,17,28,34)(11,22,12,18)(20,30,35,23)(25,36,26,31)(27,33,29,32);(1,2,3,4)(5,7)(6,14)(8,19)(9,11)(10,26)(12,15)(13,21)(16,18)(17,31)(20,27)(22,24)(23,32)(25,28)(29,35)(30,33)(34,36);(5,8,13,14)(6,15,19,9)(7,18,21,22)(10,23,28,30)(11,16,12,24)(17,26,34,25)(20,32,35,33)(27,31,29,36);(5,9,13,15)(6,16,19,24)(7,11,21,12)(8,22,14,18)(10,20,28,35)(17,23,34,30)(25,29,26,27)(31,32,36,33);(5,10)(6,17)(7,20)(8,23)(9,25)(11,27)(12,29)(13,28)(14,30)(15,26)(16,31)(18,32)(19,34)(21,35)(22,33)(24,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481193);