#217414

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

556 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,23)(22,24)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(16,21)(19,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(3,8)(5,7)(9,11,13,19,12,18,20,24)(10,15,16,17,14,22,21,23)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(2,6)(5,7)(10,14)(15,22)(16,21)(17,23)(27,33)(30,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,d^c=di

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, 124610);