#146407

order	512
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#379
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10714
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

873 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=di,f^b=fi,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, 53603);