#493247

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#442
prime radical	#3414	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

697 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=ch,e^a=eh,g^a=gi,h^a=hi,e^b=ehi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#493247, #72283942, #72285481, …
maximal normal subgroup of	C₂×#493247, #72283942, #72285481, …
maximal quotient of	C₈×AGL(1,ℤ/16), #72223577, C₂×#493247, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400443);