#406022

order	512
nilpotency class	3
isoclinism	isoclinic class of #406022
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14922
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 313218);