#465980

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 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)(3,6)(4,8)(5,7)(9,10,15,20)(11,21,25,30)(12,26,29,33)(13,19,14,18)(16,22,32,24)(17,27,34,28)(23,31,37,40)(35,38,39,36);(1,3,5,4)(2,6,7,8)(9,11,13,24,15,25,14,22)(10,16,18,21,20,32,19,30)(12,23,28,38,29,37,27,36)(17,31,26,35,34,40,33,39);(2,7)(6,8)(11,22)(12,27)(13,14)(16,30)(17,33)(18,19)(21,32)(23,37)(24,25)(26,34)(28,29)(31,40);(1,4,5,3)(2,8,7,6)(9,12)(10,17)(11,23)(13,28)(14,27)(15,29)(16,31)(18,26)(19,33)(20,34)(21,35)(22,36)(24,38)(25,37)(30,39)(32,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=f,e^2=i,g^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,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi

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, 373176);