#139855

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(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);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(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);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(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=dfh,c^2=hi,e^2=i,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, 47051);