#153023

order	512
nilpotency class	5
isoclinism	isoclinic class of #150157
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

244 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eh,d^2=ghi,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,h^a=hi,d^b=di,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60219);