#419668

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1284 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,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, 326864);