#190748

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
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

847 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)(9,10,15,20,14,19,13,18)(11,17,21,24,12,16,22,23)(27,33)(28,35)(30,37)(32,39)(34,36)(38,40);(2,5)(6,8)(9,11,14,12)(10,16,19,17)(13,22,15,21)(18,24,20,23)(25,26)(27,30)(28,32)(29,31)(33,37)(34,38)(35,39)(36,40);(9,12,14,11)(10,17,19,16)(13,22,15,21)(18,24,20,23)(25,27,29,34)(26,30,31,38)(28,36,35,33)(32,40,39,37);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,18,19,20)(11,21,12,22)(16,23,17,24)(26,31)(30,38)(32,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97944);