#134101

order	512
nilpotency class	4
isoclinism	isoclinic class of #133883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#537	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

858 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 86 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)(11,19,22,33)(12,17)(13,26,28,16)(14,18)(15,20,31,23)(21,38)(24,35)(25,34,40,32)(27,36)(29,39)(30,37);(2,5)(3,7)(9,11)(10,15)(12,23)(13,27)(14,22)(16,29)(17,33)(18,31)(19,35)(20,24)(21,28)(25,37)(26,30)(32,36)(34,38)(39,40);(10,16,18,26)(12,24)(13,28)(15,29,31,30)(17,34,35,32)(19,36,33,38)(20,23)(21,27)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41297);