#134605

order	512
nilpotency class	4
isoclinism	isoclinic class of #134444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,4,6,3,5,7,8)(9,10,14,18,24,17,23,21,15,19,22,16,13,12,11,20);(2,5)(6,8)(9,11,13,22,15,23,24,14)(10,16,17,20,19,18,12,21);(2,5)(4,7)(9,12)(10,13)(11,21)(14,18)(15,17)(16,22)(19,24)(20,23)

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,d^2=i,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=eh,f^c=fgh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41801);