#440770

order	512
nilpotency class	3
isoclinism	isoclinic class of #419445
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

1133 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 347966);