#168463

order	512
nilpotency class	3
isoclinism	isoclinic class of #168113
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

685 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,22,21)(19,23,27,26)(20,24,29,28)(25,31,32,30)(33,34,38,41)(35,43,44,47)(36,39,37,42)(40,45,48,46);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,27)(18,23,21,26)(20,25,29,32)(24,30,28,31)(34,39)(36,37)(40,47)(41,42)(43,48)(45,46);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20)(18,24)(19,25)(21,28)(22,29)(23,30)(26,31)(27,32)(33,35)(34,40)(36,46)(37,45)(38,44)(39,47)(41,48)(42,43);(2,7)(6,15)(8,12)(9,16)(17,21,22,18)(19,26,27,23)(20,28,29,24)(25,31,32,30)(34,41)(35,44)(39,42)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75659);