Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1968 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)(8,9)(10,12)(15,19)(18,20)(21,22,26,25)(23,29,31,27)(24,32,33,28)(30,34,36,35);(5,6,8,9)(7,10,12,11)(14,17)(18,20)(24,33)(28,32)(30,36)(34,35);(6,9)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,31)(22,27,25,29)(24,30,33,36)(28,34,32,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(21,24,26,33)(22,28,25,32)(23,30,31,36)(27,34,29,35);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,25,26,22)(23,29,31,27)(24,32,33,28)(30,35,36,34)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fg,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4364659);