#178878

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
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

584 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)(8,15)(9,13)(10,11)(14,16)(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)(3,8)(4,7)(5,13)(6,9)(10,15)(11,14)(12,16)(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);(1,3,6,12)(2,8,9,16)(4,11,5,10)(7,14,13,15)(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);(1,4,6,5)(2,7,9,13)(3,10,12,11)(8,15,16,14)(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=gh,b^2=i,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dgh,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, 86074);