#139159

order	512
nilpotency class	4
isoclinism	isoclinic class of #139156
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	#19335	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

315 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46355);