#134054

order	512
nilpotency class	4
isoclinism	isoclinic class of #133847
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×2₋¹⁺⁴	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

1344 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 90 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)(11,21,24,35)(12,25)(13,18)(14,19,29,33)(15,20)(16,22,32,27)(17,26,34,28)(23,36)(30,39)(31,38)(37,40);(2,5)(3,7)(9,11)(10,16)(12,22)(13,27)(14,23)(15,24)(17,30)(18,35)(19,31)(20,32)(21,25)(26,37)(28,38)(29,39)(33,40)(34,36);(9,12,15,13)(10,17,20,34)(11,22,24,27)(14,28,29,26)(16,30,32,36)(18,33,25,19)(21,31,35,40)(23,38,39,37)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fgi,e^c=ei,f^c=fhi,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, 41250);