#404214

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1134 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)(4,5)(7,8)(9,10)(11,15)(12,23)(13,16)(14,19)(17,25)(18,27)(20,34)(21,28)(22,31)(24,33)(26,30)(29,36)(32,38)(35,40)(37,39);(2,5)(6,8)(9,11,14,22)(10,15,19,31)(12,20,25,36)(13,26,27,21)(16,30,18,28)(17,29,23,34)(24,37,38,35)(32,40,33,39);(1,3,4,7)(2,6,5,8)(10,16)(13,27)(15,28)(17,32)(18,19)(21,26)(23,33)(24,38)(29,39)(30,31)(34,40)(35,37);(9,12)(10,17,19,23)(11,20)(13,24)(14,25)(15,29,31,34)(16,32,18,33)(21,35)(22,36)(26,37)(27,38)(28,39,30,40)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,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, 311410);