#475215

order	512
nilpotency class	3
isoclinism	isoclinic class of #460462
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17031
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

570 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)(3,14,18,40)(4,10)(5,11)(6,12)(7,29)(8,13)(9,17,32,19)(15,43,46,33)(16,47,50,42)(20,52,45,30)(21,34)(22,35)(23,55)(24,36)(25,37)(26,38)(27,41)(28,39)(31,49,58,51)(44,59,63,57)(48,56,64,62)(53,61)(54,60);(1,3,7,17,8,18,6,19)(2,9,29,14,13,32,12,40)(4,20,23,15,24,45,22,46)(5,16,26,51,28,50,27,49)(10,33,55,52,36,43,35,30)(11,31,38,42,39,58,41,47)(21,48,25,44,54,64,53,63)(34,59,37,62,60,57,61,56);(1,4,6,22,8,24,7,23)(2,10,12,35,13,36,29,55)(3,15,17,45,18,46,19,20)(5,25,26,54,28,53,27,21)(9,30,14,33,32,52,40,43)(11,37,38,60,39,61,41,34)(16,48,49,63,50,64,51,44)(31,57,47,62,58,59,42,56);(1,5)(2,11,13,39)(3,16)(4,21)(6,27)(7,26)(8,28)(9,31,32,58)(10,34,36,60)(12,41,29,38)(14,42,40,47)(15,44)(17,51)(18,50)(19,49)(20,48)(22,53)(23,25)(24,54)(30,56,52,62)(33,59,43,57)(35,61,55,37)(45,64)(46,63)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=e,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,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, 382411);