#138057

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
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	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

555 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 84 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)(11,23,28,52,29,47,27,51)(12,30,34,18,35,45,33,46)(13,36,24,40,37,19,38,48)(17,39,42,59,43,53,49,26)(25,55,58,50,31,62,57,71)(32,65,63,44,66,61,54,67)(41,56,69,72,70,64,60,68);(1,2,6,8,4,7,3,5)(9,11,24,53)(10,17,40,52)(12,31,63,72)(13,26,15,27)(14,28,37,59)(16,29,38,39)(18,41,65,62)(19,47,20,49)(21,42,36,23)(22,43,48,51)(25,54,68,35)(30,60,44,71)(32,64,34,58)(33,57,66,56)(45,69,67,50)(46,70,61,55);(2,7)(5,8)(9,12,16,35)(10,18,22,46)(11,25,29,31)(13,32,37,66)(14,34,15,33)(17,41,43,70)(19,44,36,67)(20,45,21,30)(23,50,47,71)(24,54,38,63)(26,56,59,64)(27,58,28,57)(39,68,53,72)(40,61,48,65)(42,60,49,69)(51,62,52,55)

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,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,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, 45253);