#178784

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

750 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,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, 85980);