#131719

order	512
nilpotency class	4
isoclinism	isoclinic class of #131654
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62252	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

589 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh,h^b=hi,e^c=egi,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38915);