#141532

order	512
nilpotency class	4
isoclinism	isoclinic class of #141454
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

283 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=ghi,e^2=ghi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,c^b=ci,d^b=dh,f^b=fh

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, 48728);