Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 33554432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2002 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)(7,11)(9,12)(13,14,16,18)(15,19,20,17)(24,31)(28,33)(29,35)(34,36);(5,6,8,10)(7,9,11,12)(13,14)(15,17)(16,18)(19,20)(22,26)(27,32)(28,33)(34,36);(1,2,3,4)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,7,8,11)(6,9,10,12)(13,15,16,20)(14,17,18,19)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(21,24,25,31)(22,28,26,33)(23,29,30,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215813);