#148972

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	#11257
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

435 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fh,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, 56168);