#138001

order	512
nilpotency class	4
isoclinism	isoclinic class of #137944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

571 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 96 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)(9,10,15,21,16,22,14,20)(11,23,28,53,29,47,27,52)(12,30,33,18,34,45,32,46)(13,35,24,40,37,19,38,49)(17,39,42,62,43,54,50,26)(25,56,59,51,60,71,58,68)(31,65,55,44,36,64,66,48)(41,61,69,72,70,57,63,67);(1,2,6,8,4,7,3,5)(9,11,24,54)(10,17,40,53)(12,25,55,72)(13,26,15,27)(14,28,37,62)(16,29,38,39)(18,41,65,56)(19,47,20,50)(21,42,35,23)(22,43,49,52)(30,63,44,51)(31,57,33,58)(32,59,36,61)(34,60,66,67)(45,69,48,68)(46,70,64,71);(9,12,16,34)(10,18,22,46)(11,25,29,60)(13,36,37,31)(14,33,15,32)(17,41,43,70)(19,48,35,44)(20,45,21,30)(23,51,47,68)(24,55,38,66)(26,61,62,57)(27,59,28,58)(39,67,54,72)(40,65,49,64)(42,63,50,69)(52,71,53,56)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,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, 45197);