Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1834 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)(6,7)(8,12)(13,14)(15,16)(17,18)(19,20)(21,22,25,26)(23,27,30,33)(24,28,32,34)(29,35,31,36);(5,6,9,7)(8,10,12,11)(14,18)(15,20)(22,26)(27,33)(28,34)(35,36);(5,6,9,7)(8,11,12,10)(13,15)(14,19)(16,18)(17,20)(21,23,25,30)(22,27,26,33)(24,31,32,29)(28,36,34,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,36)(32,34);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,20,18,15)(21,24,25,32)(22,28,26,34)(23,29,30,31)(27,35,33,36)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2014798);