#472242

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16815
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

760 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=gh,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,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, 379438);