#190803

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

971 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)(9,10,14,15,13,17,11,18)(12,20,19,23,21,16,22,24)(25,26)(27,32)(28,37)(29,34)(30,39)(31,35)(33,38)(36,40);(2,5)(6,8)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27,31,28)(26,32,35,37)(29,38,30,36)(33,39,40,34);(9,12,13,21)(10,16,17,20)(11,19,14,22)(15,23,18,24)(25,28,31,27)(26,33,35,40)(29,38,30,36)(32,39,37,34);(1,3,4,7)(2,6,5,8)(25,29,31,30)(26,34,35,39)(27,36,28,38)(32,40,37,33)

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,f^a=fh,d^b=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97999);