#241628

order	512
nilpotency class	3
isoclinism	isoclinic class of #233789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148824);