#398688

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1021 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=f,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,h^a=hi,c^b=cg,g^b=gi,g^c=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, 305884);