#6343774

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	#62781	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3116 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)(9,10,14,18,15,19,13,27)(11,20,22,30,23,16,28,31)(12,17,26,33,24,32,25,37)(21,34,36,39,35,29,38,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,20)(24,35)(25,36)(26,38)(27,31)(32,34)(33,39)(37,40);(9,12)(10,17)(11,21)(13,25)(14,26)(15,24)(16,29)(18,33)(19,32)(20,34)(22,36)(23,35)(27,37)(28,38)(30,39)(31,40);(12,24)(17,32)(21,35)(25,26)(29,34)(33,37)(36,38)(39,40);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,24,26)(16,30,20,31)(17,33,32,37)(21,36,35,38)(29,39,34,40)

Polycyclic presentation:

2^9|a^2=g,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, 6250970);