#149115

order	512
nilpotency class	4
isoclinism	isoclinic class of #140091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1918
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
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)(3,14,18,36)(4,20,24,52)(5,26)(6,30)(7,33)(8,13)(9,16,37,17)(10,22,40,23)(11,29)(12,28)(15,46)(19,51,34,42)(21,45)(25,58,56,53)(27,38,31,44)(32,43)(35,54)(39,48)(41,47,60,57)(49,62)(50,61)(55,63)(59,64);(1,3,7,34,8,18,28,19)(2,9,12,44,13,37,33,38)(4,21,25,59,24,48,56,55)(5,27,29,16,6,31,32,17)(10,39,41,64,40,45,60,63)(11,42,26,14,43,51,30,36)(15,47,54,22,50,57,49,23)(20,35,53,46,52,62,58,61);(1,4,5,22,7,25,29,57,8,24,6,23,28,56,32,47)(2,10,11,20,12,41,26,53,13,40,43,52,33,60,30,58)(3,15,16,21,19,49,27,55,18,50,17,48,34,54,31,59)(9,35,14,45,38,61,42,64,37,62,36,39,44,46,51,63)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,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, 56311);