Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 8388608 |
| 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 | |
2042 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)(6,7)(10,11)(15,18)(16,20)(21,22,25,28)(23,26,30,34)(24,31,32,27)(29,35,36,33);(5,6,9,7)(8,10,12,11)(13,14,17,19)(15,16,18,20)(21,23,25,30)(22,26,28,34)(24,29,32,36)(27,33,31,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,17,18)(14,16,19,20)(21,23)(22,26)(24,29)(25,30)(27,33)(28,34)(31,35)(32,36);(5,8,9,12)(6,10,7,11)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(21,24,25,32)(22,27,28,31)(23,29,30,36)(26,33,34,35)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215569);