#414109

order	512
nilpotency class	3
isoclinism	isoclinic class of #414109
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)(4,17)(5,6)(9,13)(10,25)(11,20)(15,16)(18,19)(23,31)(27,29)(28,32);(1,2)(3,13)(4,10)(5,20)(6,8)(7,11)(9,16)(12,24)(14,28)(15,21)(17,22)(18,26)(19,25)(23,30)(27,32)(29,31);(1,3,5,15,7,16,6,12)(2,9,8,21,11,13,20,24)(4,14,18,29,19,30,17,27)(10,23,22,31,25,28,26,32);(1,4,6,17,7,19,5,18)(2,10,8,22,11,25,20,26)(3,14,12,27,16,30,15,29)(9,23,21,31,13,28,24,32)

Polycyclic presentation:

2^9|c^2=e,d^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,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, 321305);