#6555154

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 ◁ #2298 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=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, 6462350);