#478112

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385308);