#454107

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×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2196 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)(11,21,25,30)(12,26,28,33)(13,19,14,17)(16,24,32,23)(18,29,34,27)(22,31,37,40)(35,38,39,36);(9,11,15,25)(10,16,20,32)(12,22,28,37)(13,24,14,23)(17,30,19,21)(18,31,34,40)(26,35,33,39)(27,38,29,36);(2,4)(10,17)(12,27)(13,14)(16,30)(19,20)(21,32)(22,36)(23,24)(26,33)(28,29)(31,40)(37,38);(6,8)(9,12)(10,18)(11,22)(13,29)(14,27)(15,28)(16,31)(17,33)(19,26)(20,34)(21,35)(23,38)(24,36)(25,37)(30,39)(32,40)

Polycyclic presentation:

2^9|a^2=i,b^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, 361303);