#141407

order	512
nilpotency class	4
isoclinism	isoclinic class of #141343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,10,7,9,5,3)(4,12,14,16,15,11,13,8)(17,18,21,26,23,27,22,37)(19,28,30,43,32,44,31,47)(20,33,34,39,36,48,35,25)(24,38,40,46,41,45,42,29);(1,3,6,9,7,10,5,2)(4,11,14,16,15,12,13,8)(17,19,20,29,21,30,34,38,23,32,36,46,22,31,35,45)(18,24,39,44,37,42,33,43,27,41,25,28,26,40,48,47);(1,4,6,14,7,15,5,13)(2,8,3,11,9,16,10,12)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=eh,d^2=ghi,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=dh,f^b=fh

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, 48603);