#404408

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1181 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:

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

Polycyclic presentation:

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

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, 311604);