#139117

order	512
nilpotency class	4
isoclinism	isoclinic class of #139080
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33780	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=ehi,f^c=fghi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46313);