#6547669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=gh,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^c=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6454865);