Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1996 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)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(28,33)(30,35);(6,9)(7,11)(15,19)(17,20)(23,29)(27,32)(30,35)(34,36);(6,9)(10,12)(13,15)(14,17)(16,19)(18,20)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=fh,c^2=fg,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4406969);