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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1776 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)(5,6,8,10)(7,9,11,12)(15,19)(18,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,36)(32,34)(33,35);(5,7,8,11)(6,9,10,12)(14,17)(18,20)(22,27)(23,29)(25,33)(32,36);(6,10)(9,12)(13,14,16,17)(15,18,19,20)(21,23,26,29)(22,28,27,24)(25,34,33,31)(30,32,35,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(21,24,26,28)(22,29,27,23)(25,32,33,36)(30,34,35,31);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25)(22,30)(23,31)(24,32)(26,33)(27,35)(28,36)(29,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=f,e^2=fh,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, 5387347);