#106062

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

738 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2)(3,4)(5,6,8,12,9,10,16,18)(7,13,14,20,15,11,17,19)(21,22)(23,28,33,43)(24,35,36,46)(25,31)(26,40)(27,30)(29,37,45,39)(32,47)(34,42,38,44)(41,49)(48,52)(50,51);(2,4)(6,10)(8,16)(11,13)(14,17)(21,23,26,34,27,33,25,38)(22,28,31,44,30,43,40,42)(24,32,37,49,36,50,39,48)(29,41,46,47,45,52,35,51);(5,7,9,15)(6,11,10,13)(8,17,16,14)(12,20,18,19)(21,24,27,36)(22,29,30,45)(23,32,33,50)(25,39,26,37)(28,41,43,52)(31,46,40,35)(34,49,38,48)(42,51,44,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13258);