#187368

order	512
nilpotency class	3
isoclinism	isoclinic class of #187186
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62755	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2034 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:

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

Polycyclic presentation:

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

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, 94564);