#148957

order	512
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2075
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11256
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

439 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)(3,15,19,34,20,39,17,9)(4,21,23,36,24,10,31,38)(6,12,26,43,30,44,29,40)(16,46,49,32,48,45,47,62)(18,41,27,59,52,64,50,33)(22,55,57,35,56,63,51,53)(25,54,28,60,58,37,61,42);(1,3)(2,9)(4,22,24,56)(5,19)(6,27)(7,17)(8,20)(10,35,21,53)(11,39)(12,41)(13,15)(14,34)(16,25,48,58)(18,29)(23,57,31,51)(26,52)(28,47,61,49)(30,50)(32,37,62,54)(33,40)(36,55,38,63)(42,46,60,45)(43,59)(44,64);(1,4,5,23,8,24,7,31)(2,10,11,36,14,21,13,38)(3,16,17,47,20,48,19,49)(6,28,26,58,30,61,29,25)(9,32,15,45,34,62,39,46)(12,42,40,37,44,60,43,54)(18,51,50,56,52,57,27,22)(33,55,41,35,59,63,64,53)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,f^b=fh,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, 56153);