#134196

order	512
nilpotency class	4
isoclinism	isoclinic class of #133918
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2296 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 653 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)(11,18,22,36)(12,23,26,35)(13,17)(14,19)(15,21,31,39)(16,25)(20,33)(24,40,27,29)(28,30)(32,38)(34,37);(2,5)(3,7)(9,11,14,22)(10,15)(12,20,26,25)(13,21,28,39)(16,29)(17,18)(19,31)(23,32)(24,37,27,38)(30,36)(33,40)(34,35);(9,12)(10,16,19,33)(11,20)(13,27)(14,26)(15,29,31,40)(17,34,30,32)(18,35,36,23)(21,38)(22,25)(24,28)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,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, 41392);