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 | |
2368 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)(23,28,29,35)(24,32,34,25)(30,31,33,36);(5,7,10,9)(6,11,8,12)(15,20)(17,19)(22,27)(23,29)(25,32)(30,33);(5,6,10,8)(7,12,9,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,28)(24,33)(25,31)(26,29)(27,35)(30,34)(32,36);(1,2,3,4)(5,8,10,6)(7,11,9,12)(13,15,16,20)(14,17,18,19)(21,24)(22,25)(23,30)(26,34)(27,32)(28,36)(29,33)(31,35);(5,9,10,7)(6,11,8,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,24)(23,31)(26,32)(27,34)(28,33)(29,36)(30,35)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fh,d^2=fi,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, 1621352);