Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
2130 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)(13,14,16,17)(15,18,19,20)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35);(1,2)(3,4)(6,9)(10,12)(14,17)(18,20)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863680);