Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
2038 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)(13,14,19,20)(15,16,17,18)(21,22,26,31)(23,28,32,27)(24,33,35,29)(25,34,36,30);(6,9)(7,12)(13,15)(14,18)(16,20)(17,19)(21,23,26,32)(22,27,31,28)(24,30,35,34)(25,33,36,29);(5,6,8,9)(7,10,12,11)(13,16,19,18)(14,15,20,17)(21,23)(22,28)(24,34)(25,29)(26,32)(27,31)(30,35)(33,36);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,17,19,15)(14,16,20,18)(21,24,26,35)(22,29,31,33)(23,30,32,34)(25,28,36,27);(5,7,8,12)(6,10,9,11)(13,18,19,16)(14,15,20,17)(21,25,26,36)(22,30,31,34)(23,29,32,33)(24,28,35,27)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=g,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4406742);