Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2302 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)(5,6,8,10)(7,9,11,12)(13,14,17,16)(15,18,19,20)(22,26)(27,29)(28,32)(34,35);(2,4)(15,19)(18,20)(21,22,25,26)(23,27,31,29)(24,32,33,28)(30,35,36,34);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,6)(7,9)(8,10)(11,12)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(5,7,8,11)(6,9,10,12)(13,16,17,14)(15,20,19,18)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 617018);