Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
1910 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)(5,6,9,8)(7,10,11,12)(13,14,17,15)(16,18,20,19)(21,22)(23,26)(24,32)(25,29)(27,31)(28,33)(30,35)(34,36);(6,8)(10,12)(13,15,17,14)(16,18,20,19)(21,23,25,31)(22,26,29,27)(24,30,33,36)(28,34,32,35);(5,7)(6,10)(8,12)(9,11)(13,16,17,20)(14,18,15,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,35)(32,34)(33,36);(1,2)(3,4)(5,8,9,6)(7,10,11,12)(21,22)(23,27)(24,28)(25,29)(26,31)(30,35)(32,33)(34,36);(5,7)(6,10)(8,12)(9,11)(21,24,25,33)(22,28,29,32)(23,30,31,36)(26,34,27,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2644418);