#138000

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 635 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) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 45196);