#412623

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15132
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi

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, 319819);