#96667

order	512
nilpotency class	2
isoclinism	isoclinic class of #94871
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#376
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=chi,f^a=fg,c^b=ch,e^b=eg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3863);