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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2144 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)(7,11,12,10)(14,15)(16,20)(21,22,26,30)(23,25,29,27)(24,28,33,35)(31,34,36,32);(5,6)(7,10)(8,9)(11,12)(13,14,17,15)(16,18,20,19)(21,23,26,29)(22,27,30,25)(24,31,33,36)(28,32,35,34);(6,9)(10,11)(13,14,17,15)(16,19,20,18)(21,24)(22,28)(23,31)(25,34)(26,33)(27,32)(29,36)(30,35);(1,2,3,4)(5,7,8,12)(6,10,9,11)(13,15,17,14)(16,18,20,19)(21,23)(22,27)(24,31)(25,30)(26,29)(28,32)(33,36)(34,35);(5,6)(7,11)(8,9)(10,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,27)(22,29,30,23)(24,32,33,34)(28,31,35,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=gh,d^2=fgi,e^2=fg,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, 3375204);