#493596

order	512
nilpotency class	4
isoclinism	isoclinic class of #493594
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

2316 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)(4,18,21,9)(5,10)(6,25,26,11)(7,12)(8,16,29,43)(14,36,41,55)(15,37)(17,38)(19,45,48,30)(20,31)(22,50)(23,52,53,33)(24,34)(27,40)(28,42,32,51)(35,49)(39,59,46,62)(44,61)(47,57)(54,60)(56,63)(58,64);(1,3,7,17)(2,8,12,29)(4,14,21,41)(5,22,24,15)(6,16,26,43)(9,27,18,44)(10,32,34,28)(11,13,25,38)(19,46,48,39)(20,40,49,61)(23,51,53,42)(30,56,45,54)(31,55,35,36)(33,50,52,37)(47,63,64,60)(57,59,58,62);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,27)(10,30)(11,35)(12,18)(13,36)(15,39)(16,40)(17,41)(22,46)(23,47)(24,48)(25,31)(26,49)(28,54)(29,44)(32,56)(33,58)(34,45)(37,59)(38,55)(42,60)(43,61)(50,62)(51,63)(52,57)(53,64);(1,5,7,24)(2,10,12,34)(3,15,17,22)(4,19,21,48)(6,23,26,53)(8,28,29,32)(9,30,18,45)(11,33,25,52)(13,37,38,50)(14,39,41,46)(16,42,43,51)(20,47,49,64)(27,54,44,56)(31,57,35,58)(36,59,55,62)(40,60,61,63)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400792);