#441907

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6,9,11)(7,10,8,12)(13,14)(15,27)(16,31)(17,23)(18,25)(19,24)(20,26)(21,29)(22,34)(28,36)(30,37)(32,42)(33,41)(35,40)(38,43)(39,44);(2,4)(5,7,9,8)(6,10,11,12)(13,15,20,30)(14,21,26,23)(16,32,35,43)(17,19,29,18)(22,38,41,42)(24,37,25,27)(28,33,44,34)(31,36,40,39);(1,2,3,4)(5,8,9,7)(6,10,11,12)(13,16,19,34,20,35,18,33)(14,22,25,31,26,41,24,40)(15,28,17,32,30,44,29,43)(21,36,37,38,23,39,27,42);(13,17,18,15,20,29,19,30)(14,23,24,37,26,21,25,27)(16,32,33,28,35,43,34,44)(22,39,40,38,41,36,31,42)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 349103);