#149077

order	512
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2101
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:

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

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,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,h^b=hi,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, 56273);