#498340

order	512
nilpotency class	4
isoclinism	isoclinic class of #498338
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23001
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

605 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285481, #72285570, #72285921
maximal normal subgroup of	#72285481, #72285570, #72285921
maximal quotient of	#72287327, #72287516, #72287702, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405536);