Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2210 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)(7,10)(8,12)(9,11)(15,19)(18,20)(21,22,26,27)(23,30,28,24)(25,33,36,29)(31,32,34,35);(5,7,9,12)(6,10,11,8)(14,17)(18,20)(22,27)(23,28)(29,33)(31,34);(5,7)(6,8)(9,12)(10,11)(13,14,16,17)(15,18,19,20)(21,23,26,28)(22,24,27,30)(25,34,36,31)(29,35,33,32);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15,16,19)(14,18,17,20)(21,24)(22,28)(23,27)(25,35)(26,30)(29,31)(32,36)(33,34);(5,8)(6,7)(9,10)(11,12)(21,25)(22,29)(23,31)(24,32)(26,36)(27,33)(28,34)(30,35)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgh,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3252656);