#499420

order	512
nilpotency class	4
isoclinism	isoclinic class of #499210
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#498
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2084
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

571 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)(3,13,16,39)(4,9)(5,11)(6,26)(7,10)(8,15,31,23)(12,29)(14,44,47,43)(17,50,49,41)(18,53,55,63)(19,35)(20,57)(21,37)(22,34)(24,59)(25,38)(27,32,58,62)(28,60)(30,48,61,46)(33,54,64,52)(36,45)(40,51)(42,56);(1,3,7,16)(2,8,10,31)(4,18)(5,23,25,15)(6,27,29,58)(9,33)(11,39,38,13)(12,41,26,50)(14,45)(17,51,49,24)(19,52)(20,47)(21,54)(22,55)(28,48)(30,57)(32,59,62,40)(34,64)(35,63)(36,61)(37,53)(42,44)(43,60)(46,56);(1,4,5,19,7,22,25,21)(2,9,11,35,10,34,38,37)(3,14,15,46,16,47,23,48)(6,28,24,20,29,56,51,45)(8,30,13,43,31,61,39,44)(12,42,40,36,26,60,59,57)(17,52,27,18,49,54,58,55)(32,63,41,33,62,53,50,64);(2,10)(8,31)(9,34)(11,38)(12,26)(13,39)(30,61)(32,62)(33,64)(35,37)(36,57)(40,59)(41,50)(42,60)(43,44)(53,63)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320498, #72320511, #72320528, …
maximal normal subgroup of	#72320498, #72320511, #72320528, …
maximal quotient of	#72322767, #72322906

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406616);