Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,11,12)(7,8,9,10)(15,19)(17,20)(21,22,25,26)(23,27,29,30)(24,28,33,34)(31,35,32,36);(5,7,11,9)(6,8,12,10)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(28,34)(31,32);(5,8,11,10)(6,7,12,9)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,32)(25,29)(26,30)(28,36)(31,33)(34,35);(1,2,3,4)(5,9,11,7)(6,8,12,10)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,35)(33,34);(5,10,11,8)(6,7,12,9)(13,15)(14,17)(16,19)(18,20)(21,24,25,33)(22,28,26,34)(23,31,29,32)(27,35,30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=g,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5185878);