#149155

order	512
nilpotency class	4
isoclinism	isoclinic class of #140148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1926
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11316
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

405 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=hi,f^2=hi,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,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, 56351);