#474746

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

751 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=gi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,g^d=gi,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, 381942);