#6344124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343418
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,f^2=i,g^2=hi,b^a=bf,c^a=chi,f^a=fh,f^b=fi,g^b=ghi,e^d=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251320);