Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16250 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2416 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)(7,11)(8,9)(10,12)(13,14,18,19)(15,16,17,20)(22,24)(27,29)(28,32)(34,35);(6,8)(10,11)(14,19)(15,17)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(5,7)(6,10)(8,11)(9,12)(13,15,18,17)(14,20,19,16)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,15)(17,19)(18,20)(21,24,26,22)(23,27,31,29)(25,28,33,32)(30,35,36,34);(5,8,9,6)(7,11,12,10)(13,17,18,15)(14,20,19,16)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=h,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5518926);