#137524

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂³	#449
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

311 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=ghi,f^2=gi,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi

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, 44720);