#327005

order	512
nilpotency class	3
isoclinism	isoclinic class of #326959
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1019 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234201);