#167908

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
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

619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75104);