#493452

order	512
nilpotency class	4
isoclinism	isoclinic class of #493451
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1782
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2131 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)(4,18,19,48)(5,22,25,10)(6,11)(7,12)(8,15,30,46)(9,21,31,50)(14,40,41,63)(16,38)(17,39)(20,51)(23,49)(24,54,57,34)(26,58)(27,36)(28,59)(29,45,35,56)(32,52)(33,55)(37,43,61,53)(42,62)(44,47)(60,64);(1,3,7,17)(2,8,12,30)(4,14,19,41)(5,15,25,46)(6,26,27,16)(9,28,31,60)(10,13,22,39)(11,35,36,29)(18,47,48,62)(20,42,23,44)(21,53,50,43)(24,56,57,45)(32,61,33,37)(34,58,54,38)(40,51,63,49)(52,59,55,64);(1,4)(2,9,12,31)(3,14)(5,23)(6,21)(7,19)(8,28,30,60)(10,33,22,32)(11,18,36,48)(13,37,39,61)(15,44)(16,43)(17,41)(20,25)(24,55)(26,53)(27,50)(29,62,35,47)(34,51,54,49)(38,40,58,63)(42,46)(45,59)(52,57)(56,64);(4,19)(9,31)(14,41)(18,48)(20,23)(21,50)(28,60)(32,33)(37,61)(40,63)(42,44)(43,53)(47,62)(49,51)(52,55)(59,64)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fi,g^b=gh,d^c=di,e^c=ei,g^c=gi,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, 400648);