#411392

order	512
nilpotency class	3
isoclinism	isoclinic class of #411392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15115
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

726 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)(3,13)(4,17)(5,6)(9,14)(10,25)(15,29)(16,20)(18,19)(21,24)(22,26)(23,31);(1,2,5,12,7,8,6,11)(3,14,13,21,16,24,20,9)(4,10,17,26,19,22,18,25)(15,28,27,31,29,32,30,23);(1,3)(2,9,8,21)(4,15)(5,20)(6,13)(7,16)(10,23,22,31)(11,14,12,24)(17,30)(18,27)(19,29)(25,28,26,32);(1,4,6,18,7,19,5,17)(2,10,11,25,8,22,12,26)(3,15,13,27,16,29,20,30)(9,23,14,28,21,31,24,32)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,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, 318588);