#474457

order	512
nilpotency class	3
isoclinism	isoclinic class of #464719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

721 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,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, 381653);