#149163

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×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:

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

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=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,h^b=hi,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, 56359);