#108865

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#382
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16061);