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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1950 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)(6,8)(11,12)(15,19)(18,20)(21,22,26,30)(23,31,33,27)(24,32,35,28)(25,34,36,29);(5,6,10,8)(7,11,9,12)(14,17)(18,20)(21,23,26,33)(22,27,30,31)(24,34,35,29)(25,28,36,32);(5,7,10,9)(6,11,8,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,34)(25,32)(26,33)(28,36)(29,35)(30,31);(1,2,3,4)(5,8,10,6)(7,11,9,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,35)(22,28,30,32)(23,29,33,34)(25,31,36,27);(5,9,10,7)(6,11,8,12)(21,25)(22,29)(23,32)(24,31)(26,36)(27,35)(28,33)(30,34)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=ghi,e^2=fg,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, 1488692);