#582276

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

1578 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 489472);