#316456

order	512
nilpotency class	3
isoclinism	isoclinic class of #262787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

694 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)(10,13)(11,14)(15,16)(17,18)(19,23)(20,25)(21,30)(22,26)(24,27)(28,31)(29,32);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,15,16,14)(17,19,21,27,22,29,20,28)(18,23,30,24,26,32,25,31);(9,11)(10,14)(12,16)(13,15)(17,19,22,29)(18,24,26,31)(20,27,21,28)(23,25,32,30);(2,6)(5,7)(9,10,12,13)(11,14,16,15)(17,20,22,21)(18,25,26,30)(19,27,29,28)(23,24,32,31)

Polycyclic presentation:

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

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, 223652);