#6496385

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1752 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403581);