#5041537

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1528 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4948733);