#146415

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^b=fi,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, 53611);