#403854

order	512
nilpotency class	3
isoclinism	isoclinic class of #394813
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14883
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1117 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi,f^d=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, 311050);