#6496241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3524	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403437);