#438324

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,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, 345520);