#494936

order	512
nilpotency class	4
isoclinism	isoclinic class of #13156
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1936
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1936
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285749, #72285921, #72285988, …
maximal normal subgroup of	#72285749, #72285921, #72285988, …
maximal quotient of	#72323043, #72342625

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402132);