#479865

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
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

607 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)(9,10,14,28,15,18,30,19)(11,16,12,23,21,32,22,36)(13,17,29,35,27,34,26,37)(20,31,24,39,38,40,25,33);(1,3,4,7)(2,5,6,8)(9,11,14,22,15,21,30,12)(10,16,19,23,18,32,28,36)(13,25,29,38,27,24,26,20)(17,33,37,31,34,39,35,40);(9,12,14,21,15,22,30,11)(10,16,28,23,18,32,19,36)(13,25,26,38,27,24,29,20)(17,31,37,33,34,40,35,39);(2,6)(5,8)(9,13)(10,17)(11,20)(12,24)(14,29)(15,27)(16,31)(18,34)(19,37)(21,38)(22,25)(23,39)(26,30)(28,35)(32,40)(33,36)

Polycyclic presentation:

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

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, 387061);