#371247

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278443);