Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,9)(7,11)(15,19)(18,20)(21,22,26,29)(23,30,27,24)(25,33,34,28)(31,32,35,36);(5,6,8,9)(7,10,11,12)(14,17)(18,20)(23,27)(24,30)(31,35)(32,36);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4369724);