#187874

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	#62823	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2013 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)(3,13,16,8)(4,17,19,24)(5,10)(6,12)(7,11)(9,20,25,18)(14,28)(15,26,21,23)(22,29)(27,31)(30,32)(33,34)(35,43,46,56)(36,47,49,40)(37,41)(38,42)(39,45,54,50)(44,59)(48,57,51,55)(52,58)(53,60)(61,64)(62,63);(1,3,7,16)(2,8,11,13)(4,14,19,30)(5,15,6,21)(9,22,25,27)(10,23,12,26)(17,28,24,32)(18,29,20,31)(33,35)(34,39)(36,44)(37,50)(38,46)(40,53)(41,56)(42,54)(43,58)(45,52)(47,63)(48,62)(49,61)(51,60)(55,59)(57,64);(1,4)(2,9)(3,14)(5,20)(6,18)(7,19)(8,22)(10,17)(11,25)(12,24)(13,27)(15,31)(16,30)(21,29)(23,28)(26,32)(33,36)(34,40)(35,44)(37,51)(38,49)(39,53)(41,57)(42,47)(43,59)(45,62)(46,61)(48,52)(50,60)(54,63)(55,58)(56,64);(1,5,7,6)(2,10,11,12)(3,15,16,21)(4,18,19,20)(8,23,13,26)(9,24,25,17)(14,29,30,31)(22,32,27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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, 95070);