#6344119

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343418
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

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	#62812	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3867 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 6251315);