#6481090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,33,30,34,43,49)(27,35,37,45,38,48,42,31)(28,39,41,55,40,32,46,47)(36,51,53,44,52,50,56,54);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,38)(26,31,34,45)(28,36,40,52)(29,42,43,37)(32,44,39,54)(33,48,49,35)(41,56,46,53)(47,51,55,50);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,30,40)(26,32,34,39)(27,36,38,52)(29,41,43,46)(31,44,45,54)(33,47,49,55)(35,50,48,51)(37,53,42,56);(11,18)(14,22)(17,19)(21,23)(28,40)(32,39)(36,52)(41,46)(44,54)(47,55)(50,51)(53,56);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(28,40)(32,39)(36,52)(41,46)(44,54)(47,55)(50,51)(53,56)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei

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, 6388286);