#180744

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
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

625 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,22,25)(19,26,29,32)(20,24,21,30)(23,27,31,28)(33,34,39,42)(35,45,36,47)(37,44,38,43)(40,48,41,46);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,29)(18,23,25,31)(20,28,21,27)(24,32,30,26)(33,35,39,36)(34,40,42,41)(37,48,38,46)(43,47,44,45);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(20,21)(24,30)(26,32)(27,28)(33,36,39,35)(34,41,42,40)(37,48,38,46)(43,47,44,45);(2,7)(6,15)(8,12)(9,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)(34,42)(35,36)(43,44)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87940);