#140764

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
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×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,8)(4,13)(5,15)(6,10)(7,11)(9,14)(12,16)(17,18,22,28,24,27,42,44,23,29,41,32,21,40,31,37)(19,30,26,46,34,20,36,39,35,38,47,48,33,25,43,45);(1,3,5,11,6,12,14,4)(2,7,9,8,10,13,15,16)(17,19,31,43,21,33,41,47,23,35,42,36,24,34,22,26)(18,25,44,46,40,38,28,45,29,20,37,48,27,30,32,39);(1,4,14,12,6,11,5,3)(2,8,15,7,10,16,9,13)(17,20,24,38,23,25,21,30)(18,26,40,43,29,47,27,36)(19,32,33,44,35,28,34,37)(22,39,42,48,41,45,31,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47960);