#135398

order	512
nilpotency class	4
isoclinism	isoclinic class of #124937
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

332 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,6)(4,5)(7,8)(9,10,13,17,14,20,12,26,15,18,28,35,30,19,27,36)(11,21,23,31,24,32,38,40,25,16,29,33,39,34,22,37);(2,5)(6,8)(9,11)(10,16)(12,22)(13,29)(14,24)(15,25)(17,31)(18,21)(19,32)(20,34)(23,28)(26,40)(27,38)(30,39)(33,35)(36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,17,18,35)(11,22)(13,14)(15,27)(16,31,21,33)(19,26,20,36)(23,39)(24,29)(25,38)(28,30)(32,40,34,37)

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,f^b=fi,e^c=eh,f^c=fgh,g^c=gi,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, 42594);