Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1766 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)(5,6,9,10)(7,8,11,12)(21,22,26,29)(23,30,28,25)(24,32,34,27)(31,35,36,33);(6,10)(8,12)(14,17)(18,20)(21,23,26,28)(22,25,29,30)(24,31,34,36)(27,35,32,33);(5,7,9,11)(6,8,10,12)(13,14,16,17)(15,18,19,20)(21,23)(22,25)(24,31)(26,28)(27,35)(29,30)(32,33)(34,36);(1,2,3,4)(5,8,9,12)(6,7,10,11)(21,24,26,34)(22,27,29,32)(23,31,28,36)(25,35,30,33);(5,7)(6,8)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25)(22,28)(23,29)(24,33)(26,30)(27,31)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3381934);