#140931

order	512
nilpotency class	4
isoclinism	isoclinic class of #140881
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

447 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5)(3,6)(7,8)(9,10,15,20,16,21,14,35)(11,22,26,49,27,45,36,48)(12,28,31,44,32,59,30,18)(13,33,23,39,34,19,37,46)(17,38,41,56,42,50,65,25)(24,52,54,47,55,69,62,68)(29,60,51,64,61,58,63,43)(40,53,57,66,67,71,72,70);(1,2,6,8,4,7,3,5)(9,11,23,50)(10,17,39,49)(12,24,51,70)(13,25,15,36)(14,26,34,56)(16,27,37,38)(18,40,58,69)(19,45,35,65)(20,41,33,22)(21,42,46,48)(28,57,64,47)(29,53,31,62)(30,54,61,71)(32,55,63,66)(43,68,59,72)(44,67,60,52);(9,12,15,31,16,32,14,30)(10,18,35,59,21,44,20,28)(11,24,36,62,27,55,26,54)(13,29,37,63,34,61,23,51)(17,40,65,72,42,67,41,57)(19,43,46,60,33,64,39,58)(22,47,49,69,45,68,48,52)(25,53,38,66,56,71,50,70)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48127);