#180001

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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

637 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18,22,21)(19,23,27,26)(20,24,29,28)(25,31,32,30)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,19,22,27)(18,23,21,26)(20,25,29,32)(24,30,28,31)(33,35,38,34)(36,42,46,40)(37,41,39,43)(44,45,48,47);(1,3,6,12)(2,8,10,16)(4,13,5,11)(7,14,9,15)(17,20)(18,24)(19,25)(21,28)(22,29)(23,30)(26,31)(27,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,4,6,5)(2,7,10,9)(3,11,12,13)(8,15,16,14)(17,21,22,18)(19,26,27,23)(20,28,29,24)(25,31,32,30)(33,37,38,39)(34,41,35,43)(36,45,46,47)(40,48,42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87197);