#469861

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

855 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:

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

Polycyclic presentation:

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

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, 377057);