#588486

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fg,d^2=f,e^2=fh,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, 495682);