#400326

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

935 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,14)(13,28)(15,19)(16,20)(17,24)(18,30)(22,25)(23,32)(26,27)(29,34)(31,33)(35,38)(36,39)(37,40);(1,3,5,4)(2,6,8,7)(9,11,14,24,16,26,15,25)(10,17,19,21,20,22,12,27)(13,23,29,37,31,39,30,38)(18,32,33,35,34,36,28,40);(1,3,5,4)(2,7,8,6)(9,12,14,10,16,19,15,20)(11,22,24,27,26,17,25,21)(13,28,29,18,31,33,30,34)(23,36,37,40,39,32,38,35);(1,4,5,3)(2,6,8,7)(9,13)(10,18)(11,23)(12,28)(14,29)(15,30)(16,31)(17,32)(19,33)(20,34)(21,35)(22,36)(24,37)(25,38)(26,39)(27,40)

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=ef,e^2=i,f^2=i,g^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,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307522);