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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16500 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2428 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)(8,10)(9,12)(14,18)(15,20)(21,22)(23,28)(24,30)(25,29)(26,27)(31,33)(32,36)(34,35);(5,6)(7,11)(8,10)(9,12)(13,14,17,18)(15,19,20,16)(22,27)(23,30)(29,35)(31,36);(5,6,8,10)(7,9,12,11)(13,15)(14,16)(17,20)(18,19)(21,23,26,30)(22,28,27,24)(25,31,34,36)(29,33,35,32);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,28)(22,23,27,30)(25,33,34,32)(29,36,35,31);(5,7,8,12)(6,9,10,11)(13,16,17,19)(14,20,18,15)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,33)(30,36)Polycyclic presentation:
2^9|b^2=fgh,c^2=fg,d^2=gh,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, 5524137);