#449350

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
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	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(1,2,5,4)(3,6,8,7)(9,12,13,10)(11,14,16,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30);(1,4,5,2)(3,6,8,7)(11,16)(14,15)(19,21)(23,30)(26,28)(27,31)

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=e,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356546);