#139157

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	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 304 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) = 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)(19,29,33,47)(20,35,37,25)(21,38)(22,43)(23,27)(24,32,46,39)(26,41)(28,44)(30,31)(34,36,48,40)(42,45);(1,3)(2,6,8,15)(4,11)(5,10)(7,13,9,14)(12,16)(17,19,30,35,44,48,42,32,23,33,41,25,22,34,21,39)(18,24,45,40,43,20,31,29,27,46,38,36,28,37,26,47);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,31)(21,40)(22,24)(23,37)(26,33)(27,35)(28,32)(29,30)(34,45)(36,42)(38,48)(39,43)(41,47)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=hi,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, 46353);