#471069

order	512
nilpotency class	3
isoclinism	isoclinic class of #471069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16741
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

758 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378265);