#474968

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

834 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)(11,22)(12,17)(13,16)(14,20)(15,21)(18,31)(19,23)(24,36)(25,34)(26,33)(27,35)(28,38)(29,39)(30,40)(32,37);(1,3,4,7)(2,5,6,8)(9,11,23,31,15,25,14,13)(10,16,20,34,21,18,19,22)(12,26,32,24,29,30,28,27)(17,35,38,40,39,36,37,33);(9,12,15,29)(10,17,21,39)(11,24,25,27)(13,30,31,26)(14,32,23,28)(16,33,18,40)(19,38,20,37)(22,35,34,36);(2,6)(5,8)(9,13,14,25,15,31,23,11)(10,18,20,22,21,16,19,34)(12,27,28,30,29,24,32,26)(17,36,38,33,39,35,37,40)

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,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, 382164);