#475516

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

746 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=h,d^2=e,e^2=i,g^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^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382712);