#456414

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1226 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi

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, 363610);