#187733

order	512
nilpotency class	3
isoclinism	isoclinic class of #187667
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3568	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2058 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94929);