#217077

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124273);