#479801

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=ef,f^2=i,g^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,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 386997);