Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,9)(7,12,8,11)(21,22,26,28)(23,29,30,24)(25,33,34,27)(31,32,36,35);(5,7,10,8)(6,11,9,12)(14,17)(18,20)(23,30)(24,29)(31,36)(32,35);(5,7,10,8)(6,12,9,11)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,31)(26,30)(27,32)(28,29)(33,35)(34,36);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,9,10,6)(7,11,8,12)(13,15,16,19)(14,18,17,20)(21,25,26,34)(22,27,28,33)(23,31,30,36)(24,32,29,35)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2720760);