#166966

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

670 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)(22,28)(29,32)(33,34,38,42)(35,43,44,47)(36,41,37,40)(39,46,48,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,18,21,24)(19,22,26,28)(20,23,27,30)(25,29,31,32)(33,35,38,44)(34,39,42,48)(36,46,37,45)(40,47,41,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19,21,26)(18,22,24,28)(20,25,27,31)(23,29,30,32)(34,40)(36,37)(39,47)(41,42)(43,48)(45,46);(3,12)(7,16)(10,15)(11,14)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)(35,44)(39,48)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=hi,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, 74162);