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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1726 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)(5,6)(7,12)(8,9)(10,11)(13,14,18,16)(15,19,17,20)(21,22,26,27)(23,28,31,35)(24,32,34,25)(29,30,33,36);(5,7)(6,9)(8,10)(11,12)(13,15,18,17)(14,19,16,20)(22,27)(25,32)(28,35)(30,36);(6,11)(7,8)(13,16,18,14)(15,19,17,20)(21,23)(22,28)(24,33)(25,30)(26,31)(27,35)(29,34)(32,36);(1,2,3,4)(5,8,10,7)(6,12,11,9)(13,17,18,15)(14,19,16,20)(21,24)(22,25)(23,29)(26,34)(27,32)(28,36)(30,35)(31,33);(5,9,10,12)(6,7,11,8)(21,25)(22,24)(23,30)(26,32)(27,34)(28,33)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=fgi,e^2=g,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, 1587138);