#180734

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dhi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87930);