#179404

order	512
nilpotency class	3
isoclinism	isoclinic class of #179229
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

598 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86600);