#454106

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2242 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)(3,4)(5,6)(7,8)(9,10,15,20)(11,21,25,33)(12,26,30,31)(13,16,14,19)(17,24,34,23)(18,28,36,29)(22,38,40,32)(27,35,39,37);(10,16)(13,14)(17,21)(18,31)(19,20)(23,24)(26,36)(27,39)(28,29)(32,37)(33,34)(35,38);(2,4)(9,11,15,25)(10,17,20,34)(12,27,30,39)(13,24,14,23)(16,21,19,33)(18,35,36,37)(22,29,40,28)(26,38,31,32);(6,8)(9,12)(10,18)(11,22)(13,29)(14,28)(15,30)(16,31)(17,32)(19,26)(20,36)(21,37)(23,39)(24,27)(25,40)(33,35)(34,38)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 361302);