#6344106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343406
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

2949 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)(9,10,14,20,15,19,12,17)(11,21,22,34,24,16,28,29)(13,18,27,33,26,32,25,31)(23,35,36,40,37,30,38,39);(1,3)(2,6)(4,8)(5,7)(9,11,15,24)(10,16,19,21)(12,22,14,28)(13,23,26,37)(17,29,20,34)(18,30,32,35)(25,36,27,38)(31,39,33,40);(1,4,6,5)(2,7,3,8)(9,12,15,14)(10,17,19,20)(11,22,24,28)(13,25,26,27)(16,29,21,34)(18,31,32,33)(23,36,37,38)(30,39,35,40);(9,13)(10,18)(11,23)(12,25)(14,27)(15,26)(16,30)(17,31)(19,32)(20,33)(21,35)(22,36)(24,37)(28,38)(29,39)(34,40);(13,26)(18,32)(23,37)(25,27)(30,35)(31,33)(36,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251302);