#477795

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

883 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384991);