#149159

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2073
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11316
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,e^c=ei,f^c=fi

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, 56355);