Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(6,9)(10,12)(13,14,17,18)(15,19,20,16)(21,22)(23,29)(24,26)(25,32)(27,30)(28,34)(31,35)(33,36);(13,15,17,20)(14,16,18,19)(22,24)(23,30)(28,32)(31,33);(1,2,4,3)(14,18)(16,19)(21,23)(22,27)(24,29)(25,33)(26,30)(28,36)(31,34)(32,35);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,22)(23,29,30,27)(25,32,34,28)(31,36,33,35);(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,15,18,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,36)(30,33)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fg,d^2=fgh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4314765);