#475230

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17034
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

579 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)(3,15,18,9)(4,10)(5,25)(6,14)(7,13)(8,12)(11,27)(16,30,45,42)(17,46,47,56)(19,32,20,33)(21,51,50,34)(22,53)(23,41)(24,36)(26,39)(28,40)(29,37)(31,49,43,48)(35,54)(38,57)(44,60,63,61)(52,58,64,59)(55,62);(1,3,7,20,8,18,6,19)(2,9,13,32,12,15,14,33)(4,21,23,45,24,50,29,16)(5,17,27,49,28,47,26,48)(10,34,41,30,36,51,37,42)(11,31,40,46,39,43,25,56)(22,52,54,63,55,64,57,44)(35,60,62,58,38,61,53,59);(1,4)(2,10)(3,16,18,45)(5,22)(6,23)(7,29)(8,24)(9,30,15,42)(11,38)(12,36)(13,37)(14,41)(17,44,47,63)(19,21,20,50)(25,53)(26,54)(27,57)(28,55)(31,59,43,58)(32,34,33,51)(35,39)(40,62)(46,61,56,60)(48,52,49,64);(1,5,6,26,8,28,7,27)(2,11,13,40,12,39,14,25)(3,17,19,48,18,47,20,49)(4,22,23,54,24,55,29,57)(9,31,32,46,15,43,33,56)(10,35,37,53,36,38,41,62)(16,44,21,52,45,63,50,64)(30,58,34,60,42,59,51,61)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=f,d^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,h^c=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, 382426);