#209161

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

799 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gh,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116357);