#187871

order	512
nilpotency class	3
isoclinism	isoclinic class of #187866
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	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2009 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)(3,12,15,8)(4,16,18,25)(5,19)(6,10)(7,11)(9,17,26,21)(13,28)(14,23,20,24)(22,31)(27,29)(30,32)(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);(1,3,7,15)(2,8,11,12)(4,13,18,30)(5,14,6,20)(9,22,26,27)(10,23,19,24)(16,28,25,32)(17,29,21,31)(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)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(14,29)(15,30)(16,19)(20,31)(23,32)(24,28)(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);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,17,18,21)(8,23,12,24)(9,25,26,16)(13,29,30,31)(22,32,27,28)(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=hi,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,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, 95067);