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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2240 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)(5,6,9,12)(7,11,8,10)(14,18)(16,19)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(5,7,9,8)(6,10,12,11)(13,14,17,18)(15,19,20,16)(22,26)(23,30)(24,32)(34,35);(7,8)(10,11)(13,15,17,20)(14,19,18,16)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,36)(33,35);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,15)(14,19)(16,18)(17,20)(21,22)(23,29)(24,33)(25,26)(27,30)(28,32)(31,34)(35,36);(5,6)(7,11)(8,10)(9,12)(13,16,17,19)(14,20,18,15)(21,24,25,32)(22,28,26,33)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fh,d^2=gh,e^2=fgh,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, 2992735);