#440841

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1161 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)(9,10,12,14)(11,13,15,16)(17,18,22,26,23,20,28,29)(19,24,27,31,21,25,30,32);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,13,14,16)(17,19,23,21)(18,24,20,25)(22,27,28,30)(26,31,29,32);(1,4,5,2)(3,7,8,6)(11,15)(13,16)(17,20)(18,23)(19,25)(21,24)(22,26)(27,31)(28,29)(30,32);(3,8)(6,7)(17,21,23,19)(18,25,20,24)(22,27,28,30)(26,31,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348037);