Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
2744 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)(7,11,12,9)(14,17)(15,19)(21,22)(23,26)(24,29)(25,28)(27,32)(30,34)(31,36)(33,35);(5,6,8,10)(7,9,12,11)(15,19)(18,20)(22,23)(27,29)(28,30)(35,36);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,27,32,29)(25,30,34,28)(31,35,33,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,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, 2668552);