#589134

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,d^2=fh,e^2=gh,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, 496330);