#147365

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10921
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

703 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54561);