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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1853 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)(5,6,9,7)(8,10,12,11)(13,14,17,16)(15,18,19,20)(21,22)(23,29)(24,32)(25,28)(26,31)(27,34)(30,35)(33,36);(5,6)(7,9)(8,11)(10,12)(15,19)(18,20)(21,22,25,28)(23,26,31,29)(24,32,34,27)(30,36,33,35);(6,7)(10,11)(13,15,17,19)(14,18,16,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,36)(32,34)(33,35);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,14,17,16)(15,18,19,20)(21,23)(22,26)(24,33)(25,31)(27,36)(28,29)(30,34)(32,35);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,20,19,18)(21,24)(22,27)(23,30)(25,34)(26,35)(28,32)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fh,d^2=fgh,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, 2992744);