#6744557

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

1225 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)(27,31)(28,33)(29,32)(30,34)(35,44)(36,43)(37,45)(38,42)(39,46)(40,48)(41,47)(49,54)(50,53)(51,55)(52,56);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,29,37)(26,31,32,45)(28,36,39,51)(30,35,41,38)(33,43,46,55)(34,44,47,42)(40,49,52,50)(48,54,56,53);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,35)(28,39)(30,41)(31,42)(36,50)(37,38)(43,54)(44,45)(48,56)(49,51)(53,55);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28,29,39)(26,33,32,46)(27,36,37,51)(30,40,41,52)(31,43,45,55)(34,48,47,56)(35,49,38,50)(42,53,44,54);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,35)(30,41)(31,44)(34,47)(36,49)(37,38)(40,52)(42,45)(43,54)(48,56)(50,51)(53,55)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,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, 6651753);