#139928

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

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

normal subgroups only

This group has 507 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)(19,21,29,36)(20,30,33,42)(22,25)(23,26)(24,32,41,38)(27,44)(28,35,37,46)(31,45)(34,47)(39,43)(40,48);(1,3,5,8,6,9,15,2)(4,10,14,11,12,16,13,7)(17,19,21,26,23,29,36,18)(20,27,34,41,33,40,31,24)(22,37,35,25,39,28,46,43)(30,38,45,48,42,32,47,44);(1,4)(2,7)(3,10)(5,14)(6,12)(8,11)(9,16)(13,15)(17,20)(18,24)(19,27)(21,34)(22,38)(23,33)(25,42)(26,41)(28,47)(29,40)(30,43)(31,36)(32,39)(35,48)(37,45)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,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, 47124);