#217061

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
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

739 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)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24)(25,26)(27,32)(28,38)(29,35)(30,34)(31,36)(33,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,31,28)(26,32,36,38)(29,39,30,37)(33,34,40,35);(2,5)(6,8)(10,15)(12,19)(14,20)(18,23)(25,28,31,27)(26,33,36,40)(29,39,30,37)(32,34,38,35);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,29,31,30)(26,34,36,35)(27,37,28,39)(32,33,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=ei,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124257);