#412692

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1140 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,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,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319888);