#139515

order	512
nilpotency class	4
isoclinism	isoclinic class of #139440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,e^c=ei,f^c=fgh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46711);