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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,19,20,16)(21,22,27,30)(23,28,26,36)(24,25,34,29)(31,35,33,32);(2,4)(13,15)(14,16)(17,20)(18,19)(21,23,27,26)(22,28,30,36)(24,31,34,33)(25,35,29,32);(1,2,3,4)(14,18)(16,19)(21,24)(22,29)(23,31)(25,30)(26,33)(27,34)(28,32)(35,36);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,18)(17,20)(21,25,27,29)(22,24,30,34)(23,32,26,35)(28,33,36,31);(5,7,8,11)(6,9,10,12)(13,16,17,19)(14,20,18,15)(21,26,27,23)(22,28,30,36)(24,33,34,31)(25,35,29,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=h,d^2=fg,e^2=fi,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, 3540292);