Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,10)(7,11)(14,17)(18,20)(21,22)(23,27)(24,32)(25,26)(28,34)(29,30)(31,36)(33,35);(5,6,9,10)(7,12,11,8)(15,19)(18,20)(22,26)(23,29)(28,32)(31,33);(7,11)(8,12)(13,14,16,17)(15,18,19,20)(21,23,25,29)(22,27,26,30)(24,33,34,31)(28,36,32,35);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,34)(33,35);(5,8)(6,7)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,24)(22,28)(23,31)(25,34)(26,32)(27,35)(29,33)(30,36)Polycyclic presentation:
2^9|b^2=fh,c^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72268481 |
| maximal normal subgroup of | #72268481 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4944295);