Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4544 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)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,25)(18,32)(19,27)(21,30)(23,29)(24,28)(26,33)(31,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,17)(15,30)(16,27)(22,31)(23,33)(24,34)(25,35)(28,32)(29,36);(2,4)(5,8,10,25,11,23,9,24)(6,13,14,22,16,28,15,29)(7,18,20,35,21,33,19,34)(12,26,17,31,27,32,30,36);(2,4)(8,23)(13,28)(18,33)(22,29)(24,25)(26,32)(31,36)(34,35);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,23,25)(12,17,27,30)(13,22,28,29)(18,34,33,35)(26,31,32,36)Polycyclic presentation:
2^9|c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349779);