#147160

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#387
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10877
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

491 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54356);