#214813

order	512
nilpotency class	3
isoclinism	isoclinic class of #214704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122009);