#107309

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

764 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)(7,15,8,11)(9,14)(10,13)(12,17,19,16)(18,20)(21,22,26,31,27,32,25,30)(23,33,37,28,36,43,35,42)(24,29,38,44,39,45,40,46)(34,48,49,52,50,47,51,41);(1,2,3,4)(5,7,10,8)(6,11,13,15)(9,16,18,17)(12,14,19,20)(21,23)(22,28)(24,34)(25,37)(26,35)(27,36)(29,41)(30,43)(31,33)(32,42)(38,51)(39,50)(40,49)(44,47)(45,52)(46,48);(5,8,10,7)(6,12,13,19)(9,16,18,17)(11,14,15,20)(21,24)(22,29)(23,34)(25,40)(26,38)(27,39)(28,41)(30,46)(31,44)(32,45)(33,47)(35,51)(36,50)(37,49)(42,52)(43,48)

Polycyclic presentation:

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

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, 14505);