#466762

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ef,d^2=fi,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^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373958);