#105869

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

846 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 13065);