#406014

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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1021 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)(3,6)(4,5)(7,8)(9,10,14,18,15,20,13,19)(11,21,25,32,26,22,24,16)(12,27,29,17,30,33,28,34)(23,31,38,35,39,40,37,36);(2,5)(6,8)(9,11,14,25,15,26,13,24)(10,16,19,22,20,32,18,21)(12,23,29,38,30,39,28,37)(17,31,27,36,34,40,33,35);(9,10)(11,22)(12,17)(13,18)(14,19)(15,20)(16,24)(21,26)(23,36)(25,32)(27,29)(28,33)(30,34)(31,37)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,30,14,29)(10,17,18,33,20,34,19,27)(11,23,24,37,26,39,25,38)(16,31,21,35,32,40,22,36)

Polycyclic presentation:

2^9|a^2=f,b^2=h,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, 313210);