#109740

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,10,7,11,5,20)(3,12,15,24,16,23,14,8)(4,17,18,9,19,25,21,26)(13,28,29,32,31,22,30,27)(33,34,37,42,39,43,38,52)(35,44,47,58,46,40,48,54)(36,41,50,56,49,55,51,62)(45,53,60,63,59,57,61,64);(1,3)(2,8,11,24)(4,13)(5,14)(6,15)(7,16)(9,22,26,28)(10,12,20,23)(17,32,25,27)(18,29)(19,31)(21,30)(33,35,38,48,39,46,37,47)(34,40,42,54,43,44,52,58)(36,45,51,61,49,59,50,60)(41,53,56,63,55,57,62,64);(1,4,6,18,7,19,5,21)(2,9,10,25,11,26,20,17)(3,13,15,29,16,31,14,30)(8,22,12,27,24,28,23,32)(33,36,37,50,39,49,38,51)(34,41,42,56,43,55,52,62)(35,45,47,60,46,59,48,61)(40,53,54,63,44,57,58,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=eh

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, 16936);