#135404

order	512
nilpotency class	4
isoclinism	isoclinic class of #124708
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
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×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,6)(4,5)(7,8)(9,10,13,19,14,17,29,34,15,20,28,37,12,18,26,35)(11,21,24,38,25,16,30,32,23,31,27,40,22,33,39,36);(2,5)(6,8)(9,11,15,23)(10,16)(12,22,14,25)(13,27,28,24)(17,31)(18,21)(19,36)(20,33)(26,30,29,39)(32,37)(34,38)(35,40);(1,3,4,7)(2,6,5,8)(10,17)(12,14)(13,26)(16,31)(18,20)(19,37)(21,33)(22,25)(24,39)(27,30)(28,29)(32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=dg,d^2=i,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,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, 42600);