#406015

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

989 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)(3,6)(4,5)(7,8)(9,10,15,20,16,21,14,30)(11,22,26,37,27,36,25,17)(12,28,31,33,19,13,32,18)(23,34,35,40,39,24,29,38);(2,5)(6,8)(9,11,14,25,16,27,15,26)(10,17,20,22,21,37,30,36)(12,29,31,23,19,35,32,39)(13,24,33,40,28,34,18,38);(9,12,16,19)(10,18,21,33)(11,23,27,39)(13,30,28,20)(14,31,15,32)(17,34,37,24)(22,38,36,40)(25,35,26,29);(1,3,4,7)(2,6,5,8)(9,13,14,33,16,28,15,18)(10,19,20,32,21,12,30,31)(11,24,25,40,27,34,26,38)(17,35,22,39,37,29,36,23)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 313211);