#479774

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

606 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)(3,5)(4,6)(7,8)(9,10,13,19,14,15,22,18)(11,20,21,30,24,32,25,16)(12,17,28,34,29,31,26,35)(23,36,37,40,27,39,38,33);(1,3,4,7)(2,5,6,8)(10,15)(11,21)(12,26)(13,22)(16,32)(17,34)(20,30)(24,25)(28,29)(31,35)(33,40)(37,38);(9,11,22,25,14,24,13,21)(10,16,18,32,15,30,19,20)(12,27,28,38,29,23,26,37)(17,33,34,36,31,40,35,39);(2,6)(5,8)(9,12)(10,17)(11,23)(13,28)(14,29)(15,31)(16,33)(18,35)(19,34)(20,36)(21,37)(22,26)(24,27)(25,38)(30,40)(32,39)

Polycyclic presentation:

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

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, 386970);