#135353

order	512
nilpotency class	4
isoclinism	isoclinic class of #124850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1820
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=hi,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ci,e^c=eh,f^c=fghi,f^e=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, 42549);