#465981

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

866 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)(3,6)(4,8)(5,7)(9,10,16,12)(11,21,24,34)(13,27,30,33)(14,20,15,19)(17,31,25,23)(18,28,26,29)(22,32,38,39)(35,37,40,36);(1,3,5,4)(2,6,7,8)(9,11,14,23,16,24,15,31)(10,17,19,21,12,25,20,34)(13,22,29,37,30,38,28,36)(18,32,27,35,26,39,33,40);(2,7)(6,8)(9,12,16,10)(11,21,24,34)(13,27,30,33)(14,19,15,20)(17,31,25,23)(18,28,26,29)(22,32,38,39)(35,37,40,36);(1,4,5,3)(2,8,7,6)(9,13)(10,18)(11,22)(12,26)(14,29)(15,28)(16,30)(17,32)(19,27)(20,33)(21,35)(23,37)(24,38)(25,39)(31,36)(34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,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, 373177);