#166796

order	512
nilpotency class	3
isoclinism	isoclinic class of #166289
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

721 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)(3,10,13,16)(4,8,5,14)(7,11,15,12)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)(33,34)(35,43)(36,41)(37,40)(38,42)(39,45)(44,47)(46,48);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,28,30,31)(33,35)(34,39)(36,46)(37,45)(38,44)(40,47)(41,43)(42,48);(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)(34,40)(36,37)(39,47)(41,42)(43,48)(45,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(35,44)(39,48)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dg,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, 73992);