#178721

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	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

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,19)(18,22)(20,29)(21,27)(23,31)(24,32)(25,28)(26,30)(33,35,37,43)(34,38,40,46)(36,42,44,48)(39,45,41,47);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,15)(10,16)(17,18)(19,22)(20,28)(21,23)(24,30)(25,29)(26,32)(27,31)(34,39)(36,44)(38,45)(40,41)(42,48)(46,47);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,14,16,15)(18,23)(22,31)(24,28)(25,32)(34,40)(38,46)(39,41)(45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gh,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, 85917);