Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2202 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)(7,10)(8,11)(9,12)(14,18)(15,19)(21,22,26,27)(23,28,31,35)(24,29,34,33)(25,32,36,30);(5,7)(6,9)(8,12)(10,11)(13,14,17,18)(15,16,19,20)(22,27)(23,31)(29,33)(30,32);(1,2,3,4)(21,23)(22,28)(24,32)(25,33)(26,31)(27,35)(29,36)(30,34);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,19)(18,20)(21,24)(22,29)(23,32)(25,35)(26,34)(27,33)(28,36)(30,31);(5,6)(7,9)(8,11)(10,12)(13,16)(14,19)(15,18)(17,20)(21,25,26,36)(22,30,27,32)(23,33,31,29)(24,28,34,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4334526);