Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1967 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)(5,6,9,11)(7,8,12,10)(13,14,17,19)(15,16,20,18)(21,22)(23,28)(24,29)(25,30)(26,27)(31,32)(33,35)(34,36);(5,7,9,12)(6,10,11,8)(13,15)(14,18)(16,19)(17,20)(22,27)(25,29)(28,35)(32,36);(6,11)(7,12)(13,16,17,18)(14,15,19,20)(21,23)(22,28)(24,34)(25,32)(26,33)(27,35)(29,36)(30,31);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,24,26,30)(22,29,27,25)(23,31,33,34)(28,32,35,36);(5,8,9,10)(6,7,11,12)(13,14)(15,16)(17,19)(18,20)(21,25,26,29)(22,30,27,24)(23,32,33,36)(28,31,35,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1481733);