#180626

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

(2,7)(4,5)(8,14)(9,13)(10,12)(15,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32)(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)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(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)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)(33,36)(34,40)(35,43)(37,46)(38,47)(39,51)(41,54)(42,55)(44,58)(45,59)(48,60)(49,61)(50,62)(52,63)(53,57)(56,64);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14)(19,26)(22,30)(27,32)(29,31)(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,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, 87822);