#139163

order	512
nilpotency class	4
isoclinism	isoclinic class of #139160
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	#19337	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

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

normal subgroups only

This group has 316 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) = 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)(17,18,22,27,24,20,23,35)(19,28,32,45,33,44,40,46)(21,36,38,34,29,47,39,26)(25,41,42,31,30,48,43,37);(1,3)(2,6,8,15)(4,11)(5,10)(7,13,9,14)(12,16)(17,19,29,31,23,32,38,48,24,33,21,37,22,40,39,41)(18,25,26,44,27,43,36,45,20,30,34,28,35,42,47,46);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,24)(19,30)(21,26)(22,27)(23,35)(25,33)(28,31)(29,34)(32,42)(36,39)(37,44)(38,47)(40,43)(41,45)(46,48)

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=ehi,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi

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, 46359);