#149099

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11293
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fhi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi

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, 56295);