#190789

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
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

835 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)(11,16)(12,22)(13,19)(14,18)(15,20)(17,23)(21,24)(25,26,30,32,28,33,29,34)(27,35,38,40,36,31,37,39);(2,5)(6,8)(9,11,15,12)(10,16,20,22)(13,23,14,21)(17,18,24,19)(25,27,28,36)(26,31,33,35)(29,37,30,38)(32,39,34,40);(9,12,15,11)(10,17,20,24)(13,23,14,21)(16,18,22,19)(26,32)(29,30)(31,39)(33,34)(35,40)(37,38);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,21,12,23)(16,17,22,24)(26,33)(31,35)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=ei,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dh,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, 97985);