Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22,26,30)(23,27,28,24)(25,33,34,29)(31,32,36,35);(6,9)(10,12)(13,15,18,16)(14,19,17,20)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,35)(34,36);(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,19,17,20)(21,23)(22,27)(24,30)(25,31)(26,28)(29,35)(32,33)(34,36);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,17,18,14)(15,19,16,20)(21,24,26,27)(22,28,30,23)(25,32,34,35)(29,36,33,31);(21,25,26,34)(22,29,30,33)(23,31,28,36)(24,32,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,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, 854302);