#403798

order	512
nilpotency class	3
isoclinism	isoclinic class of #394813
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14882
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

942 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,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,h^a=hi,c^b=cg,f^b=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, 310994);