Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2264 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)(6,9)(7,11)(14,17)(15,19)(21,22,27,24)(23,25,31,28)(26,29,35,32)(30,33,36,34);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,27,31)(22,28,24,25)(26,30,35,36)(29,34,32,33);(5,6,8,9)(7,10,11,12)(13,15)(14,18)(16,19)(17,20)(21,24,27,22)(23,28,31,25)(26,32,35,29)(30,34,36,33);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,14)(15,20)(16,17)(18,19)(21,25,27,28)(22,23,24,31)(26,34,35,33)(29,36,32,30);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,19)(17,20)(21,26)(22,29)(23,30)(24,32)(25,33)(27,35)(28,34)(31,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2646076);