#403031

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1267 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^c=ei,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, 310227);