#416538

order	512
nilpotency class	3
isoclinism	isoclinic class of #397240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fh,c^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi

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, 323734);