Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19258 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
464 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)(11,20)(12,17)(13,26)(14,29)(15,19)(16,23)(18,27)(21,34)(22,35)(24,37)(25,36)(28,30)(31,39)(32,38)(33,40);(9,11,21,38,14,22,13,23,15,24,27,39,12,25,28,40)(10,16,30,35,17,32,18,20,19,33,26,36,29,31,34,37);(2,5)(3,8)(6,7);(2,6)(5,7)(9,12,15,14)(10,17,19,29)(11,22,24,25)(13,21,28,27)(16,31,33,32)(18,26,34,30)(20,35,37,36)(23,39,40,38)Polycyclic presentation:
2^9|b^2=egi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,d^b=di,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309639 |
| maximal normal subgroup of | #72309639 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398308);