#459923

order	512
nilpotency class	3
isoclinism	isoclinic class of #459923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16352
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

931 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=f,d^2=f,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,d^c=de,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, 367119);