#426218

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

939 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=fi,d^2=e,e^2=i,f^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=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi

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, 333414);