#215238

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

896 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)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22)(27,33)(28,35)(30,37)(32,39)(34,36)(38,40);(1,3,4,8)(2,5,6,7)(25,26)(27,30)(28,32)(29,31)(33,37)(34,38)(35,39)(36,40);(3,8)(5,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,29,34)(26,30,31,38)(28,36,35,33)(32,40,39,37);(2,6)(5,7)(26,31)(30,38)(32,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,f^c=fi,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, 122434);