#107615

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14811);