Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2081 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)(14,18)(15,20)(21,22,26,27)(23,28,31,35)(24,32,34,25)(29,30,33,36);(5,7,10,12)(6,9,8,11)(13,14)(15,19)(16,20)(17,18)(22,27)(25,32)(28,35)(30,36);(6,8)(7,12)(13,15,17,20)(14,19,18,16)(21,23)(22,28)(24,33)(25,30)(26,31)(27,35)(29,34)(32,36);(1,2)(3,4)(5,8,10,6)(7,9,12,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,25)(23,29)(26,34)(27,32)(28,36)(30,35)(31,33);(5,9)(6,7)(8,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,25)(22,24)(23,30)(26,32)(27,34)(28,33)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4376853);