#460615

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

952 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,18,14)(12,16,26,23)(13,17)(21,25,36,31)(22,30)(24,32)(27,37)(28,35)(29,34)(33,40)(38,39);(1,3,4,7)(2,5,6,8)(9,11,14,10,15,18,20,19)(12,23,25,36,26,16,31,21)(13,22,30,17,29,35,28,34)(24,38,40,33,39,32,27,37);(2,6)(5,8)(9,12,15,26)(10,16,19,23)(11,21,18,36)(13,27,29,40)(14,31,20,25)(17,33,34,37)(22,32,35,38)(24,30,39,28);(9,13)(10,17)(11,22)(12,24)(14,30)(15,29)(16,32)(18,35)(19,34)(20,28)(21,37)(23,38)(25,40)(26,39)(27,31)(33,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=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, 367811);