#493564

order	512
nilpotency class	4
isoclinism	isoclinic class of #493562
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1790
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62886	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2320 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)(3,13)(4,9)(5,10)(6,24,25,11)(7,12)(8,16,28,46)(14,39)(15,37)(17,38)(18,29)(19,35,48,30)(20,31)(21,32)(22,50,51,33)(23,34)(26,41,53,62)(27,44,55,64)(36,42)(40,59)(43,58)(45,60)(47,57,49,56)(52,61,54,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,26)(10,27)(11,13)(12,28)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,38)(25,46)(29,52)(30,39)(31,53)(32,54)(33,37)(34,55)(35,36)(47,61)(48,62)(49,63)(50,60)(51,64)(56,59)(57,58);(1,4,7,20)(2,9,12,31)(3,14,17,42)(5,21,23,18)(6,19,25,48)(8,26,28,53)(10,32,34,29)(11,35,24,30)(13,36,38,39)(15,43,45,40)(16,41,46,62)(22,49,51,47)(27,54,55,52)(33,56,50,57)(37,59,60,58)(44,63,64,61);(1,5,7,23)(2,10,12,34)(3,15,17,45)(4,18,20,21)(6,22,25,51)(8,27,28,55)(9,29,31,32)(11,33,24,50)(13,37,38,60)(14,40,42,43)(16,44,46,64)(19,47,48,49)(26,52,53,54)(30,56,35,57)(36,58,39,59)(41,61,62,63)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256593, #72256596, #72256597, …
maximal normal subgroup of	#72256593, #72256596, #72256597, …
maximal quotient of	#72256615, #72256654

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400760);