#499558

order	512
nilpotency class	4
isoclinism	isoclinic class of #498968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#486
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2099
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

527 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,e^2=i,g^2=hi,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=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320498, #72320615, #72320632, …
maximal normal subgroup of	#72320498, #72320615, #72320632, …
maximal quotient of	#72322793, #72323067

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406754);