#6372233

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279429);