#179220

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

637 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=hi,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, 86416);