#451027

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
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

1381 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)(9,10,13,19,14,16,15,29)(11,20,23,30,21,34,24,17)(12,25,28,31,27,33,26,18)(22,37,39,40,36,32,38,35);(2,5)(3,7)(10,16)(11,21)(13,15)(17,30)(22,36)(25,33)(26,28)(35,40);(9,11,14,21)(10,17,16,30)(12,22,27,36)(13,24,15,23)(18,32,31,37)(19,34,29,20)(25,40,33,35)(26,39,28,38);(1,3,4,7)(2,6,5,8)(9,12)(10,18,16,31)(11,22)(13,28)(14,27)(15,26)(17,32,30,37)(19,25,29,33)(20,35,34,40)(21,36)(23,39)(24,38)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,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, 358223);