#6682039

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26)(27,35)(28,40)(29,33)(30,34)(31,38)(32,41)(36,46)(37,45)(39,44)(42,53)(43,48)(47,54)(49,52)(50,55)(51,56);(1,3,4,7)(2,5,6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27,29,38)(26,31,33,35)(28,37,41,52)(30,36,43,39)(32,45,40,49)(34,44,48,46)(42,50,54,51)(47,55,53,56);(2,6)(5,8)(27,36)(28,41)(30,43)(31,44)(32,40)(34,48)(35,46)(37,51)(38,39)(45,56)(49,55)(50,52);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,29,41)(26,32,33,40)(27,37,38,52)(30,42,43,54)(31,45,35,49)(34,47,48,53)(36,50,39,51)(44,55,46,56);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(26,33)(27,36)(30,43)(31,46)(32,40)(35,44)(37,50)(38,39)(42,54)(45,56)(49,55)(51,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,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, 6589235);