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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2346 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)(7,11)(8,10)(9,12)(13,14,16,18)(15,19,20,17)(21,22,26,27)(23,25,31,28)(24,32,33,29)(30,36,34,35);(5,7,10,9)(6,11,8,12)(15,20)(17,19)(22,27)(24,33)(25,28)(30,34);(5,6,10,8)(7,12,9,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,28)(24,34)(25,27)(26,31)(29,35)(30,33)(32,36);(1,2,3,4)(5,8,10,6)(7,11,9,12)(13,15,16,20)(14,17,18,19)(21,24,26,33)(22,29,27,32)(23,30,31,34)(25,35,28,36);(5,9,10,7)(6,11,8,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,28)(22,23,27,31)(24,35,33,36)(29,30,32,34)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=i,e^2=fh,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, 1621345);