#451032

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1388 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)(3,6)(4,5)(7,8)(9,10,15,19,13,17,14,27)(11,20,28,34,21,16,23,31)(12,24,29,32,25,39,26,18)(22,36,40,30,35,38,37,33);(2,5)(3,7)(9,11)(10,16)(12,22)(13,21)(14,28)(15,23)(17,20)(18,33)(19,34)(24,38)(25,35)(26,40)(27,31)(29,37)(30,32)(36,39);(10,17)(11,21)(14,15)(16,20)(18,32)(22,35)(26,29)(30,33);(1,3,4,7)(2,6,5,8)(9,12,13,25)(10,18)(11,22,21,35)(14,26,15,29)(16,30)(17,32)(19,24)(20,33)(23,37,28,40)(27,39)(31,38)(34,36)

Polycyclic presentation:

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

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, 358228);