#479858

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

616 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=gh,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 387054);