#509856

order	512
nilpotency class	4
isoclinism	isoclinic class of #509855
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31064
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

803 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defh,g^a=gi,c^b=cg,d^b=de,e^b=eh,f^b=fi,h^b=hi,f^c=fi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417052);