#137801

order	512
nilpotency class	4
isoclinism	isoclinic class of #137768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

512 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44997);