Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2215 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)(5,6,8,9)(7,10,11,12)(13,14)(15,16)(17,19)(18,20)(21,22,26,30)(23,31,33,27)(24,34,35,28)(25,32,36,29);(6,9)(10,12)(13,15,18,19)(14,17,20,16)(21,23,26,33)(22,27,30,31)(24,32,35,29)(25,28,36,34);(7,11)(10,12)(13,16,18,17)(14,15,20,19)(21,23)(22,27)(24,32)(25,34)(26,33)(28,36)(29,35)(30,31);(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(13,17,18,16)(14,19,20,15)(21,24,26,35)(22,28,30,34)(23,32,33,29)(25,27,36,31);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,19)(18,20)(21,25,26,36)(22,29,30,32)(23,28,33,34)(24,31,35,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3375186);