#434762

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341958);