Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470237 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,15,21,28)(12,16,23,30)(13,24,25,17)(19,31,33,26)(20,34,35,27)(22,36,37,29)(32,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,26)(17,27)(18,28)(22,32)(23,33)(24,34)(25,35)(29,38)(30,31)(36,39)(37,40);(3,7)(5,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,29)(18,30)(20,32)(21,33)(24,36)(25,37)(27,38)(28,31)(34,39)(35,40);(3,7)(5,8)(9,13,14,25)(10,17,18,24)(11,20,21,35)(12,22,23,37)(15,27,28,34)(16,29,30,36)(19,32,33,40)(26,38,31,39);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=i,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260824, #72260844, #72303498, … |
| maximal normal subgroup of | #72260824, #72260844, #72303498, … |
| maximal quotient of | #72260933, #72261097 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377435);