#148730

order	512
nilpotency class	4
isoclinism	isoclinic class of #13460
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2058
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11207
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

501 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=gi,h^c=hi

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, 55926);