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) = 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 | |
3294 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)(21,22,26,30)(23,24,28,27)(25,33,34,29)(31,35,36,32);(1,2,3,4)(6,9)(10,12)(14,17)(18,20)(23,28)(24,27)(25,34)(29,33);(13,14)(15,18)(16,17)(19,20)(21,23,26,28)(22,27,30,24)(25,31,34,36)(29,35,33,32);(5,6)(7,10)(8,9)(11,12)(21,24,26,27)(22,28,30,23)(25,32,34,35)(29,36,33,31);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,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, 6038978);