#314784

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

612 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:

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

Polycyclic presentation:

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

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, 221980);