Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #31046 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22)(23,26)(24,32)(25,30)(27,29)(28,33)(31,35)(34,36);(1,2)(3,4)(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,26,22)(24,28,34,31)(25,30,27,29)(32,35,36,33);(2,4)(21,24,26,34)(22,28,23,31)(25,33,27,35)(29,32,30,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(22,23)(28,31)(29,30)(32,36);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,26,27)(22,29,23,30)(24,33,34,35)(28,32,31,36)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584934);