#140929

order	512
nilpotency class	4
isoclinism	isoclinic class of #140881
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

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

normal subgroups only

This group has 455 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)(11,21)(12,26)(13,31,34,18)(14,35)(15,20)(16,23,38,32)(17,29)(19,27)(22,45)(24,51)(25,44)(28,57,60,53)(30,54)(33,41,56,62)(36,65,66,72)(37,63,64,50)(39,59,61,71)(40,55)(42,48)(43,46)(47,68,58,52)(49,69)(67,70);(1,2,6,8,4,7,3,5)(9,11,13,32)(10,16,31,44)(12,22,28,58)(14,24,33,63)(15,25,34,23)(17,36,61,67)(18,21,20,38)(19,37,62,51)(26,52,57,69)(27,46,56,50)(29,48,59,72)(30,49,60,47)(35,64,41,43)(39,42,40,66)(45,54,68,53)(55,70,71,65);(9,12,27,55,15,30,14,29)(10,17,35,54,20,40,19,26)(11,22,46,70,25,49,24,48)(13,28,56,71,34,60,33,59)(16,36,64,68,38,66,37,52)(18,39,62,57,31,61,41,53)(21,42,51,69,44,67,43,45)(23,47,63,72,32,58,50,65)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=ghi,e^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, 48125);