#6681990

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 6589186);