Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(7,11)(13,14,17,20)(15,16,19,18)(21,22,26,24)(23,27,31,29)(25,28,34,32)(30,35,33,36);(1,2,3,4)(6,9)(10,12)(15,19)(16,18)(22,24)(27,29)(28,32)(35,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,20,16)(21,23,26,31)(22,27,24,29)(25,33,34,30)(28,36,32,35);(5,6)(7,10)(8,9)(11,12)(13,16,17,18)(14,19,20,15)(21,24,26,22)(23,29,31,27)(25,28,34,32)(30,35,33,36);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=fgi,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3454862);