#180752

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

603 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:

(1,2)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31)(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);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32)(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)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29)(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);(2,8)(3,10)(6,9)(11,16)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,27,32,31)(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=hi,c^2=hi,d^2=h,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, 87948);