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 | |
2041 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)(7,12)(8,9)(10,11)(13,14,17,19)(15,16,20,18)(21,22,26,27)(23,25,29,35)(24,31,34,28)(30,36,32,33);(5,7)(6,9)(8,10)(11,12)(13,15,17,20)(14,18,19,16)(22,27)(25,35)(28,31)(33,36);(6,11)(7,8)(14,19)(16,18)(21,23,26,29)(22,25,27,35)(24,32,34,30)(28,36,31,33);(1,2,3,4)(5,8,10,7)(6,12,11,9)(13,16,17,18)(14,15,19,20)(21,24,26,34)(22,28,27,31)(23,30,29,32)(25,33,35,36);(5,9,10,12)(6,7,11,8)(13,15)(14,18)(16,19)(17,20)(21,25)(22,29)(23,27)(24,33)(26,35)(28,32)(30,31)(34,36)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fg,d^2=i,e^2=h,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, 1587979);