#212749

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

589 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)(9,10,14,18)(11,16,20,24)(12,21,19,15)(13,23,22,17)(25,26,30,33,31,34,29,38)(27,35,36,39,37,40,28,32);(2,5)(6,8)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24)(25,27,29,28,31,37,30,36)(26,32,33,35,34,39,38,40);(1,3,4,7)(2,6,5,8)(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)(25,27,30,36,31,37,29,28)(26,32,38,40,34,39,33,35);(25,28,29,37,31,36,30,27)(26,32,38,40,34,39,33,35)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=eh,d^2=gh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119945);