#326919

order	512
nilpotency class	3
isoclinism	isoclinic class of #326869
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

871 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,28)(21,30)(22,25)(24,29)(26,31)(27,32);(2,5)(6,8)(9,10,12,16)(11,15,13,14)(17,19,22,20)(18,23,25,28)(21,29,31,27)(24,26,32,30);(1,3)(2,6)(4,7)(5,8)(10,14)(11,13)(15,16)(17,20,22,19)(18,24,25,32)(21,27,31,29)(23,30,28,26);(3,7)(6,8)(9,11,12,13)(10,15,16,14)(18,25)(23,28)(24,32)(26,30)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234115);