#434667

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

1090 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,c^2=f,d^2=i,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,f^b=fi,g^b=gi,e^d=ei

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, 341863);