#413608

order	512
nilpotency class	3
isoclinism	isoclinic class of #403107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2547
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15150
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

895 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,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, 320804);