#474970

order	512
nilpotency class	3
isoclinism	isoclinic class of #464719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382166);