Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6,7,8)(9,10,13,18,14,17,26,33)(11,19,21,34,22,15,27,29)(12,16,24,32,25,31,37,40)(20,30,35,39,23,28,36,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,17,19)(12,23,25,20)(13,21,26,27)(16,30,31,28)(18,34,33,29)(24,36,37,35)(32,38,40,39);(1,3,5,4)(2,6,7,8);(9,12,14,25)(10,16,17,31)(11,20,22,23)(13,24,26,37)(15,28,19,30)(18,32,33,40)(21,35,27,36)(29,38,34,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^e=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388738);