#456626

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 363822);