#404663

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14898
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 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)(3,13,17,15)(4,18,22,27)(5,6)(7,11)(8,12)(9,16,25,23)(10,20,28,21)(14,24)(19,26)(29,30)(31,32);(1,3,7,16,8,17,6,23)(2,9,11,13,12,25,5,15)(4,19,20,14,22,30,21,31)(10,26,18,24,28,29,27,32);(1,4,6,20,8,22,7,21)(2,10,5,18,12,28,11,27)(3,14,16,19,17,31,23,30)(9,24,13,26,25,32,15,29);(1,5,7,2,8,11,6,12)(3,15,16,9,17,13,23,25)(4,18,21,10,22,27,20,28)(14,29,19,24,31,26,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311859);