#139519

order	512
nilpotency class	4
isoclinism	isoclinic class of #139440
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	#25734	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 628 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)(9,10,14,19)(11,20,25,37)(12,17)(13,18,29,33)(15,22)(16,28)(21,35)(23,32)(24,30,27,40)(26,34)(31,38)(36,39);(1,3,5,2)(4,7,6,8)(9,11,21,23,14,25,12,22)(10,15,17,20,19,32,35,37)(13,27,28,38,29,24,26,39)(16,30,18,36,34,40,33,31);(4,6)(7,8)(9,12,14,21)(10,16,19,34)(11,22,25,23)(13,28,29,26)(15,30,32,40)(17,18,35,33)(20,36,37,31)(24,39,27,38)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=i,f^2=ghi,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, 46715);