#140980

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

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	#127
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

427 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48176);