#498125

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	#33540	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

713 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^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, #72285988
maximal normal subgroup of	#72285481, #72285570, #72285988
maximal quotient of	#72287350, #72287525, #72287706, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405321);