#187876

order	512
nilpotency class	3
isoclinism	isoclinic class of #187872
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	#62825	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

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, 95072);