Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1652 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)(7,9)(8,10)(11,12)(13,14,18,20)(15,17,19,16)(21,22,26,29)(23,30,28,25)(24,32,35,27)(31,34,33,36);(7,11)(9,12)(13,15,18,19)(14,16,20,17)(23,28)(25,30)(31,33)(34,36);(5,6,8,10)(7,9,11,12)(13,15)(14,16)(17,20)(18,19)(21,23,26,28)(22,25,29,30)(24,33,35,31)(27,36,32,34);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,17)(14,15,20,19)(21,24,26,35)(22,27,29,32)(23,31,28,33)(25,34,30,36);(5,7,8,11)(6,9,10,12)(13,17,18,16)(14,19,20,15)(21,25)(22,28)(23,29)(24,34)(26,30)(27,33)(31,32)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=fi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5417241);