#498127

order	512
nilpotency class	4
isoclinism	isoclinic class of #498124
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#22858
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3584	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

719 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=eh,g^b=gi,h^b=hi,f^c=fi,e^d=ehi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285481, #72285570, #72285725
maximal normal subgroup of	#72285481, #72285570, #72285725
maximal quotient of	#72287279, #72287493, #72287685, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405323);