#478244

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=f,d^2=ef,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, 385440);