#414105

order	512
nilpotency class	3
isoclinism	isoclinic class of #414105
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15162
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

971 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=gh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,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, 321301);