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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2211 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)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,20,16)(21,22,25,26)(23,29,31,27)(24,32,33,28)(30,35,34,36);(5,6,8,10)(7,9,11,12)(13,15,17,20)(14,16,18,19)(22,26)(24,33)(27,29)(30,34);(5,7,8,11)(6,9,10,12)(13,14,17,18)(15,19,20,16)(21,23)(22,27)(24,34)(25,31)(26,29)(28,36)(30,33)(32,35);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,24,25,33)(22,28,26,32)(23,30,31,34)(27,35,29,36);(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,33)(34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fg,d^2=hi,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, 1620092);