#479758

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

764 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)(9,10,16,20,15,22,14,21)(11,23,27,35,28,34,26,17)(12,29,31,37,13,19,32,18)(24,38,30,36,40,33,25,39);(1,3,4,7)(2,5,6,8)(9,11,14,26,15,28,16,27)(10,17,20,23,22,35,21,34)(12,30,31,40,13,25,32,24)(18,36,19,38,37,39,29,33);(9,12,14,31,15,13,16,32)(10,18,20,19,22,37,21,29)(11,24,26,30,28,40,27,25)(17,33,23,36,35,38,34,39);(2,6)(5,8)(9,13)(10,19)(11,25)(12,15)(14,31)(16,32)(17,33)(18,20)(21,37)(22,29)(23,39)(24,27)(26,40)(28,30)(34,36)(35,38)

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,e^d=ei,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, 386954);