#451028

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

1365 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)(9,10,14,17,15,19,13,20)(11,21,24,32,25,16,23,30)(12,26,27,18,28,33,29,35)(22,36,37,34,38,40,39,31);(2,5)(3,7)(9,11,15,25)(10,16,19,21)(12,22,28,38)(13,24,14,23)(17,32,20,30)(18,34,35,31)(26,40,33,36)(27,39,29,37);(10,17)(13,14)(16,30)(18,33)(19,20)(21,32)(23,24)(26,35)(27,29)(31,36)(34,40)(37,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18,19,35)(11,22)(13,29)(14,27)(15,28)(16,31,21,34)(17,33,20,26)(23,39)(24,37)(25,38)(30,36,32,40)

Polycyclic presentation:

2^9|a^2=gh,b^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, 358224);