Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 4194304 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2301 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)(15,19)(18,20)(21,22,26,28)(23,29,30,25)(24,32,34,27)(31,33,36,35);(1,2,3,4)(6,9)(10,12)(14,17)(18,20)(23,30)(25,29)(31,36)(33,35);(13,14,16,17)(15,18,19,20)(21,23)(22,25)(24,31)(26,30)(27,33)(28,29)(32,35)(34,36);(5,6)(7,10)(8,9)(11,12)(21,24,26,34)(22,27,28,32)(23,31,30,36)(25,33,29,35);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,23)(24,33)(26,29)(27,31)(28,30)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039149);