#6343775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343312
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	#62793	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3104 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:

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

Polycyclic presentation:

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

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, 6250971);