#128939

order	512
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=i,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=ch,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36135);