#6444122

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351318);