#416816

order	512
nilpotency class	3
isoclinism	isoclinic class of #401626
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

851 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)(3,7)(4,5)(6,8)(9,10,14,18,15,19,13,30)(11,20,24,32,23,37,22,16)(12,25,28,35,29,17,27,34)(21,36,26,31,39,33,38,40);(2,5)(6,7)(9,11,13,22,15,23,14,24)(10,16,18,20,19,32,30,37)(12,26,27,21,29,38,28,39)(17,33,34,40,25,36,35,31);(9,12,15,29)(10,17,19,25)(11,21,23,39)(13,28,14,27)(16,31,32,40)(18,35,30,34)(20,36,37,33)(22,26,24,38);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

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, 324012);