#319616

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

959 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226812);