#139929

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#447	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47125);