#499358

order	512
nilpotency class	4
isoclinism	isoclinic class of #498968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#486
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2071
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

717 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)(14,44,46,43)(17,49,48,41)(18,51,52,64)(19,35)(20,54)(21,37)(22,34)(24,58)(25,38)(27,32,57,62)(28,60)(30,47,61,45)(33,50,63,53)(36,56)(40,59)(42,55);(1,3)(2,8)(4,18)(5,23)(6,27)(7,16)(9,33)(10,31)(11,39)(12,41)(13,38)(14,20)(15,25)(17,24)(19,53)(21,50)(22,52)(26,49)(28,47)(29,57)(30,36)(32,40)(34,63)(35,64)(37,51)(42,44)(43,60)(45,55)(46,56)(48,59)(54,61)(58,62);(1,4,5,19,7,22,25,21)(2,9,11,35,10,34,38,37)(3,14,15,45,16,46,23,47)(6,28,24,56,29,55,59,20)(8,30,13,43,31,61,39,44)(12,42,40,54,26,60,58,36)(17,50,27,52,48,53,57,18)(32,51,41,63,62,64,49,33);(2,10)(8,31)(9,34)(11,38)(12,26)(13,39)(30,61)(32,62)(33,63)(35,37)(36,54)(40,58)(41,49)(42,60)(43,44)(51,64)

Polycyclic presentation:

2^9|a^2=f,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	#72320541, #72320602, #72320615, …
maximal normal subgroup of	#72320541, #72320602, #72320615, …
maximal quotient of	#72322993, #72323062

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406554);