Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,11,9)(7,12,10,8)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,35)(33,34);(5,7,11,10)(6,12,9,8)(14,17)(18,20)(22,26)(27,29)(28,34)(35,36);(5,8)(6,10)(7,9)(11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,36)(33,34);(1,2)(3,4)(5,9,11,6)(7,12,10,8)(13,15,16,19)(14,18,17,20)(21,23)(22,27)(24,32)(25,31)(26,29)(28,36)(30,33)(34,35);(5,10,11,7)(6,12,9,8)(21,24,25,33)(22,28,26,34)(23,30,31,32)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=fgh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2518662);