Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| 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 | |
2122 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)(7,11)(10,12)(13,14,17,19)(15,20,18,16)(22,26)(23,29)(24,32)(34,36);(6,9)(10,12)(15,18)(16,20)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,16,19,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fg,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 854288);