#6740824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1178 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)(3,7)(4,8)(5,6)(9,10,14,17)(11,19,21,15)(12,16,13,18)(20,24,22,23)(25,26)(27,36)(28,40)(29,44)(30,35)(31,34)(32,39)(33,43)(37,47)(38,48)(41,50)(42,51)(45,49)(46,52)(53,55)(54,56);(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27,30,39)(26,32,35,36)(28,41,43,54)(29,38,31,45)(33,50,40,56)(34,48,44,49)(37,46,55,42)(47,51,53,52);(2,6)(7,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,42)(30,43)(31,46)(32,47)(34,51)(35,40)(36,53)(38,54)(39,55)(41,45)(44,52)(48,50)(49,56);(1,3,5,4)(2,7,6,8)(10,17)(15,19)(16,18)(23,24)(25,29,30,31)(26,34,35,44)(27,38,39,45)(28,42,43,46)(32,48,36,49)(33,51,40,52)(37,54,55,41)(47,50,53,56);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,38)(28,43)(29,31)(32,49)(33,40)(34,44)(36,48)(37,41)(39,45)(47,50)(53,56)(54,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,d^2=g,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6648020);