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 | |
1812 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)(5,6,9,7)(8,10,12,11)(13,14,17,15)(16,18,20,19)(21,22)(23,31)(24,34)(25,33)(26,27)(28,32)(29,35)(30,36);(6,7)(10,11)(13,15,17,14)(16,18,20,19)(22,27)(23,32)(29,34)(30,33);(5,6,9,7)(8,11,12,10)(13,16)(14,18)(15,19)(17,20)(21,23)(22,28)(24,33)(25,29)(26,32)(27,31)(30,35)(34,36);(1,2,3,4)(5,7,9,6)(8,10,12,11)(21,24,26,35)(22,29,27,34)(23,33,32,30)(25,31,36,28);(5,8,9,12)(6,10,7,11)(21,25)(22,30)(23,34)(24,28)(26,36)(27,33)(29,32)(31,35)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fg,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, 3373095);