#451033

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

1358 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)(9,10,13,27,14,18,30,19)(11,20,23,17,22,15,28,35)(12,24,25,34,26,16,29,36)(21,32,38,40,37,31,39,33);(2,5)(3,7)(9,11)(10,15)(12,21)(13,28)(14,22)(16,32)(17,27)(18,20)(19,35)(23,30)(24,31)(25,39)(26,37)(29,38)(33,36)(34,40);(9,12,14,26)(10,16,18,24)(11,21,22,37)(13,29,30,25)(15,31,20,32)(17,33,35,40)(19,36,27,34)(23,39,28,38);(1,3,4,7)(2,6,5,8)(9,11)(10,17,18,35)(12,21)(13,23)(14,22)(15,19,20,27)(16,33,24,40)(25,38)(26,37)(28,30)(29,39)(31,36,32,34)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=e,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, 358229);