Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,17,16,20)(22,23)(24,31)(25,33)(35,36);(2,4)(6,9)(10,11)(14,19)(17,20)(21,22,26,23)(24,29,31,27)(25,30,33,28)(32,35,34,36);(1,2,3,4)(13,15,18,16)(14,17,19,20)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,36,34,35);(5,7,8,12)(6,10,9,11)(13,16,18,15)(14,17,19,20)(21,24,26,31)(22,27,23,29)(25,34,33,32)(28,36,30,35);(5,6)(7,11)(8,9)(10,12)(13,17)(14,16)(15,19)(18,20)(21,25,26,33)(22,28,23,30)(24,32,31,34)(27,35,29,36)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2800017);