Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2298 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)(13,14)(15,20)(16,18)(17,19)(21,22,26,27)(23,28,31,33)(24,32,34,29)(25,35,36,30);(5,7)(6,9)(8,12)(10,11)(13,15,17,16)(14,18,19,20)(22,27)(23,31)(29,32)(30,35);(13,15,17,16)(14,18,19,20)(21,23)(22,28)(24,30)(25,32)(26,31)(27,33)(29,36)(34,35);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,16,17,15)(14,18,19,20)(21,24,26,34)(22,29,27,32)(23,30,31,35)(25,28,36,33);(5,6)(7,9)(8,11)(10,12)(13,14)(15,18)(16,20)(17,19)(21,25,26,36)(22,30,27,35)(23,32,31,29)(24,33,34,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gi,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, 4342389);