Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,10)(7,11,12,9)(22,24)(23,29)(28,32)(31,35);(5,6)(7,11)(8,10)(9,12)(13,14,16,18)(15,17,19,20)(21,22,26,24)(23,30,29,27)(25,28,33,32)(31,36,35,34);(7,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(21,24,26,22)(23,27,29,30)(25,28,33,32)(31,36,35,34);(5,6)(7,11)(8,10)(9,12)(13,15,16,19)(14,17,18,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=f,e^2=gh,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, 2672157);