#6343196

order	512
nilpotency class	3
isoclinism	isoclinic class of #61559
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3108 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)(11,19)(12,16)(13,25)(14,17)(15,21)(18,27)(20,33)(22,32)(23,30)(24,37)(26,29)(28,35)(31,36)(34,40)(38,39);(1,3,6,2)(4,7,5,8)(9,11,14,22)(10,15,17,29)(12,20,23,34)(13,26,27,21)(16,28,30,39)(18,32,25,19)(24,38,36,35)(31,40,37,33);(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,30)(18,31)(19,33)(21,35)(22,34)(25,37)(26,38)(27,36)(29,39)(32,40);(12,23)(16,30)(20,34)(24,36)(28,39)(31,37)(33,40)(35,38);(2,3)(7,8)(10,17)(15,29)(16,30)(18,25)(19,32)(28,39)(31,37)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=bf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,d^c=dhi

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250392);