#138396

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=gh,d^2=ghi,e^2=i,f^2=ghi,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, 45592);