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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1992 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)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,18)(17,20)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,35,36,31);(5,6,8,10)(7,9,12,11)(13,15,18,17)(14,19,16,20)(22,27)(23,28)(25,34)(31,35);(5,6)(7,9)(8,10)(11,12)(13,15,18,17)(14,20,16,19)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,35)(34,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,19,17,20)(21,24)(22,28)(23,27)(25,35)(26,32)(29,30)(31,34)(33,36);(5,7)(6,9)(8,12)(10,11)(13,17,18,15)(14,19,16,20)(21,25,26,34)(22,29,27,33)(23,30,28,36)(24,31,32,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=fgh,d^2=fgh,e^2=gh,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, 2992821);