#6372179

order	512
nilpotency class	3
isoclinism	isoclinic class of #60367
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3382 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)(11,19,23,32)(12,24)(13,18)(14,17)(15,22,30,27)(16,25)(20,28)(21,35,36,33)(26,39)(29,40,38,37)(31,34);(2,5)(6,7)(9,11,20,22,14,23,13,27)(10,15,28,19,17,30,18,32)(12,21,34,37,25,36,26,40)(16,29,39,33,24,38,31,35);(1,3,4,8)(2,6,5,7)(9,12,14,25)(10,16,17,24)(11,21,23,36)(13,26,20,34)(15,29,30,38)(18,31,28,39)(19,33,32,35)(22,37,27,40);(12,25)(16,24)(21,36)(26,34)(29,38)(31,39)(33,35)(37,40);(10,17)(15,30)(16,24)(18,28)(19,32)(29,38)(31,39)(33,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6279375);