#441910

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fi,d^2=ei,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, 349106);