#606448

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=i,e^2=fghi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 513644);