#168708

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
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

678 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75904);