Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470237 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4195 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)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,25)(14,17)(18,26)(19,30)(20,29)(21,35)(22,27)(23,37)(24,31)(28,34)(32,38)(33,39)(36,40);(3,4)(6,7)(9,11)(10,15)(12,22)(13,21)(14,20)(16,30)(17,29)(18,28)(19,24)(23,36)(25,35)(26,34)(27,31)(32,40)(33,38)(37,39);(1,3,5,4)(2,6,8,7)(9,12,14,24)(10,16,17,31)(11,19,20,22)(13,23,26,38)(15,27,29,30)(18,32,25,37)(21,33,34,36)(28,39,35,40);(1,4,5,3)(2,7,8,6)(10,17)(11,20)(16,31)(18,25)(19,22)(21,34)(32,37)(33,36);(2,8)(6,7)(9,13,14,26)(10,18,17,25)(11,21,20,34)(12,23,24,38)(15,28,29,35)(16,32,31,37)(19,33,22,36)(27,39,30,40)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^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 | #72311259, #72311272, #72311315, … |
| maximal normal subgroup of | #72311259, #72311272, #72311315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394851);