Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(6,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,30)(24,28)(25,27)(26,33)(29,34)(31,35)(32,36);(2,3)(6,10)(11,12)(14,15)(19,20)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(1,3,4,2)(5,6)(7,11)(8,12)(9,10)(13,15,18,14)(16,20,17,19)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(5,7,9,8)(6,11,10,12)(13,16,18,17)(14,19,15,20)(21,25,27,22)(23,28,24,30)(26,33,34,29)(31,35,32,36);(5,8,9,7)(6,12,10,11)(13,17,18,16)(14,20,15,19)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=f,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863669);