#139513

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	#25733	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 590 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,6,8,5,3,4,7)(9,10)(11,19,22,13)(12,23)(14,28)(15,18)(16,25)(17,30)(20,35,37,40)(21,27,39,34)(24,36,33,38)(26,32)(29,31);(1,3,5,2)(4,7,6,8)(9,11,13,18)(10,15,22,19)(12,20,24,31)(14,21,27,28)(16,26,37,33)(17,30,39,34)(23,29,38,35)(25,36,40,32);(4,6)(7,8)(9,12,15,26)(10,16,18,31)(11,20,22,37)(13,24,19,33)(14,29,30,25)(17,32,28,23)(21,38,39,36)(27,35,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,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, 46709);