Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482362 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,26,27,17)(14,18)(15,21,29,36)(20,33)(22,34)(23,37)(24,39,40,31)(25,32)(28,35,30,38);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,29)(12,23,25,20)(13,21,27,36)(16,30,32,28)(17,19,26,34)(24,38,40,35)(31,37,39,33);(2,6)(5,7)(9,12,14,25)(10,16,18,32)(11,20,22,23)(13,24,27,40)(15,28,29,30)(17,31,26,39)(19,33,34,37)(21,35,36,38);(2,6)(5,7)(9,13)(10,17)(11,21)(12,24)(14,27)(15,19)(16,31)(18,26)(20,35)(22,36)(23,38)(25,40)(28,33)(29,34)(30,37)(32,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284994, #72285068, #72285101, … |
| maximal normal subgroup of | #72284994, #72285068, #72285101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389560);