#213428

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,23,22,19)(25,26,30,33,29,32,38,28)(27,34,37,36,35,31,39,40);(1,3,5,4)(2,6,7,8)(10,14)(15,22)(16,21)(17,24)(25,27,29,35)(26,31,32,34)(28,36,33,40)(30,37,38,39);(9,11,13,19,12,18,20,23)(10,15,16,17,14,22,21,24)(27,35)(31,34)(36,40)(37,39);(1,4,5,3)(2,6,7,8)(25,28,38,32,29,33,30,26)(27,36,39,34,35,40,37,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,d^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120624);