#139151

order	512
nilpotency class	4
isoclinism	isoclinic class of #139146
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19258	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 327 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)(3,6)(4,7,12,9)(10,15)(11,13)(14,16)(17,18,21,26,23,27,22,37)(19,28,32,44,30,42,33,48)(20,34,35,41,29,45,36,25)(24,38,40,46,39,47,43,31);(1,3)(2,6,8,15)(4,11)(5,10)(7,13,9,14)(12,16)(17,19,29,46,21,32,36,47,23,30,20,31,22,33,35,38)(18,24,25,28,37,43,45,48,27,39,41,42,26,40,34,44);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(19,30)(24,39)(28,42)(31,46)(32,33)(38,47)(40,43)(44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fh,e^c=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46347);