#140493

order	512
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1947
prime radical	C₂³	#477
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ei,d^2=gi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47689);