#5227570

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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)(5,6,9,10)(7,13,14,11)(8,12,16,17)(15,19,20,18)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34)(37,38,42,45)(39,43,40,47)(41,44,49,52)(46,50,48,51);(6,10)(7,14)(8,16)(18,19)(23,29)(27,30)(31,35)(34,36)(37,39,42,40)(38,43,45,47)(41,48,49,46)(44,51,52,50);(5,6)(7,11)(8,17)(9,10)(12,16)(13,14)(15,19)(18,20)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36)(37,40,42,39)(38,43,45,47)(41,48,49,46)(44,50,52,51);(1,2)(3,4)(5,7,9,14)(6,11,10,13)(8,15,16,20)(12,18,17,19)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34)(37,38)(39,43)(40,47)(41,44)(42,45)(46,50)(48,51)(49,52);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)(37,41,42,49)(38,44,45,52)(39,46,40,48)(43,50,47,51)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=gh,d^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5134766);