#441861

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

1013 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 349057);