#434681

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=e,c^2=fi,d^2=i,e^2=i,f^2=i,g^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,g^b=gi,e^d=ei,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, 341877);