#6547741

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

1234 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)(27,36,39,48)(28,40,43,33)(29,34,30,46)(32,44,49,38)(37,54,41,53)(42,52,45,51)(47,56,50,55);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,44,31,39,30,38)(26,32,46,48,35,49,34,36)(28,41,45,56,43,37,42,55)(33,50,51,53,40,47,52,54);(2,5)(6,8)(10,15)(11,17)(16,22)(18,23)(25,28,31,43)(26,33,35,40)(27,37,39,41)(29,45,30,42)(32,47,49,50)(34,52,46,51)(36,53,48,54)(38,56,44,55);(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)(27,38,39,44)(28,42,43,45)(32,48,49,36)(33,51,40,52)(37,55,41,56)(47,53,50,54)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,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^b=ei,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, 6454937);