Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2124 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)(5,6)(7,11)(8,10)(9,12)(13,14,16,18)(15,19,20,17)(21,22)(23,24)(25,33)(26,27)(28,34)(29,32)(30,35)(31,36);(5,7,10,9)(6,11,8,12)(13,14,16,18)(15,17,20,19)(22,27)(23,29)(25,34)(31,35);(5,6,10,8)(7,12,9,11)(13,15)(14,17)(16,20)(18,19)(21,23,26,29)(22,24,27,32)(25,30,34,36)(28,31,33,35);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,35)(26,32)(27,29)(28,36)(30,33)(31,34);(5,9,10,7)(6,11,8,12)(13,15)(14,17)(16,20)(18,19)(21,25)(22,28)(23,30)(24,31)(26,34)(27,33)(29,36)(32,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=h,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2991970);