Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499892 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,14,18)(11,15,21,27)(12,16,23,29)(13,17,24,30)(19,31,33,25)(20,26,34,37)(22,28,35,38)(32,39,40,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,25)(17,26)(18,27)(22,32)(23,33)(24,34)(28,36)(29,31)(30,37)(35,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,22)(14,23)(15,25)(17,28)(18,29)(20,32)(21,33)(24,35)(26,36)(27,31)(30,38)(34,40)(37,39);(3,8)(5,7)(9,13)(10,17)(11,20)(12,22)(14,24)(15,26)(16,28)(18,30)(19,32)(21,34)(23,35)(25,36)(27,37)(29,38)(31,39)(33,40);(2,6)(5,7)(13,24)(17,30)(20,34)(22,35)(26,37)(28,38)(32,40)(36,39)Polycyclic presentation:
2^9|a^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303126, #72303134, #72303149, … |
| maximal normal subgroup of | #72303126, #72303134, #72303149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407090);