#475394

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17046
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

671 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)(3,15,19,33)(4,20)(5,11)(6,12)(7,13)(8,14)(9,18,34,30)(10,24)(16,44,46,31)(17,47,50,43)(21,35,51,60)(22,39)(23,38)(25,37)(26,57)(27,40)(28,41)(29,42)(32,49,62,52)(36,55)(45,63,58,59)(48,64,54,61)(53,56);(1,3,7,18,8,19,6,30)(2,9,13,15,14,34,12,33)(4,21,23,16,25,51,24,46)(5,17,27,52,29,50,28,49)(10,35,20,44,38,60,37,31)(11,32,40,43,42,62,41,47)(22,54,55,45,56,48,26,58)(36,61,53,63,57,64,39,59);(1,4)(2,10)(3,16)(5,26)(6,24)(7,23)(8,25)(9,31)(11,39)(12,37)(13,20)(14,38)(15,35)(17,48)(18,51)(19,46)(21,30)(22,27)(28,56)(29,55)(32,61)(33,60)(34,44)(36,40)(41,57)(42,53)(43,63)(45,49)(47,59)(50,54)(52,58)(62,64);(1,5,8,29)(2,11)(3,17,19,50)(4,22,25,56)(6,28,7,27)(9,32)(10,36)(12,41)(13,40)(14,42)(15,43)(16,45,46,58)(18,52,30,49)(20,53)(21,54,51,48)(23,55,24,26)(31,59)(33,47)(34,62)(35,61)(37,39)(38,57)(44,63)(60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382590);