#466495

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

781 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:

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

Polycyclic presentation:

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

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, 373691);