#140780

order	512
nilpotency class	3
isoclinism	isoclinic class of #113356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂³	#435
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,11)(4,7)(5,14)(6,10)(8,12)(9,15)(13,16)(17,18,22,28,24,27,38,43,23,19,25,31,21,30,37,46)(20,32,35,47,36,29,41,45,33,44,39,48,34,26,40,42);(1,3,5,4,6,12,15,13)(2,7,9,11,10,16,14,8)(17,19,22,28,24,30,38,43,23,18,25,31,21,27,37,46)(20,32,39,47,36,29,40,45,33,44,35,48,34,26,41,42);(1,4,5,12,6,13,15,3)(2,8,9,7,10,11,14,16)(17,20,21,34,23,33,24,36)(18,26,27,32,19,29,30,44)(22,35,37,40,25,39,38,41)(28,42,43,47,31,45,46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cgh,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47976);