#4467789

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1570 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)(9,16)(11,14)(12,17)(15,18)(19,20)(21,22,26,29)(23,30,27,24)(25,33,34,28)(31,32,35,36)(38,39)(40,47)(44,46)(48,52);(6,8)(7,14)(9,17)(18,19)(23,27)(24,30)(31,35)(32,36)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,7)(6,11)(8,13)(9,15)(10,14)(12,18)(16,19)(17,20)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36)(37,39,42,38)(40,43,47,45)(41,46,49,44)(48,50,52,51);(1,2)(3,4)(5,8,10,6)(7,11,14,13)(9,16,17,12)(15,18,20,19)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36)(37,40,42,47)(38,43,39,45)(41,48,49,52)(44,50,46,51);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,34)(22,28,29,33)(23,31,27,35)(24,32,30,36)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4374985);