#6669767

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,33)(23,34,36,29)(24,37,40,30)(25,41,28,31)(26,42,45,32)(35,47,46,48)(38,49,44,52)(39,50,43,51);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,27,36)(22,29,33,34)(24,38,40,44)(25,35,28,46)(26,43,45,39)(30,49,37,52)(31,47,41,48)(32,51,42,50);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,27,40)(22,30,33,37)(23,26,36,45)(25,39,28,43)(29,42,34,32)(31,50,41,51)(35,44,46,38)(47,49,48,52);(6,12)(10,13)(11,18)(17,19)(21,25,27,28)(22,31,33,41)(23,35,36,46)(24,39,40,43)(26,44,45,38)(29,47,34,48)(30,50,37,51)(32,52,42,49);(7,15)(10,13)(14,20)(17,19)(21,26,27,45)(22,32,33,42)(23,24,36,40)(25,38,28,44)(29,37,34,30)(31,49,41,52)(35,43,46,39)(47,50,48,51)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6576963);