Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(10,12)(14,15)(16,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,34)(31,35)(33,36);(2,4)(5,6,9,8)(7,10,11,12)(14,15)(18,19)(22,23)(27,29)(28,30)(35,36);(1,2,3,4)(13,14,17,15)(16,18,20,19)(21,23,26,22)(24,27,32,29)(25,30,34,28)(31,35,33,36);(5,7)(6,10)(8,12)(9,11)(13,15,17,14)(16,19,20,18)(21,24)(22,27)(23,29)(25,33)(26,32)(28,36)(30,35)(31,34);(5,8,9,6)(7,12,11,10)(13,16)(14,18)(15,19)(17,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)Polycyclic presentation:
2^9|b^2=fgi,c^2=h,d^2=fg,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,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, 2698787);