#6744569

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1216 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,29,33)(27,35,39,31)(28,40,41,32)(30,34,43,48)(36,46,38,44)(37,45,52,49)(42,53,54,47)(50,55,51,56);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,29,39)(26,31,33,35)(28,37,41,52)(30,36,43,38)(32,45,40,49)(34,44,48,46)(42,50,54,51)(47,55,53,56);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(27,36)(28,41)(30,43)(31,44)(32,40)(34,48)(35,46)(37,51)(38,39)(45,56)(49,55)(50,52);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28,29,41)(26,32,33,40)(27,37,39,52)(30,42,43,54)(31,45,35,49)(34,47,48,53)(36,50,38,51)(44,55,46,56);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,38)(30,43)(31,46)(34,48)(35,44)(36,39)(37,51)(42,54)(45,56)(47,53)(49,55)(50,52)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fg,e^2=fgi,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, 6651765);