#240896

order	512
nilpotency class	3
isoclinism	isoclinic class of #240895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10,15,20)(11,21,23,33)(12,26,29,17)(13,19,14,18)(16,25,32,24)(22,37,38,34)(27,40,39,31)(28,35,30,36);(3,8)(5,6)(9,11)(10,16)(12,27)(13,25)(14,24)(15,23)(17,34)(18,33)(19,21)(20,32)(22,30)(26,37)(28,38)(29,39)(31,36)(35,40);(2,5)(3,8)(6,7)(9,12,13,28,15,29,14,30)(10,17,18,35,20,26,19,36)(11,22,24,39,23,38,25,27)(16,31,21,37,32,40,33,34);(11,23)(16,32)(21,33)(22,38)(24,25)(27,39)(31,40)(34,37)

Polycyclic presentation:

2^9|a^2=h,c^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,g^b=gh,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148092);