#6498218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470355
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1283 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,33)(27,35,37,51)(28,32,40,48)(29,42,43,34)(31,44,46,38)(36,50,39,54)(41,56,52,49)(45,55,47,53);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,29,38,30,37,43,44)(26,31,34,35,33,46,42,51)(28,39,41,55,40,36,52,53)(32,47,49,54,48,45,56,50);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,28)(26,32)(27,36)(29,41)(30,40)(31,45)(33,48)(34,49)(35,50)(37,39)(38,53)(42,56)(43,52)(44,55)(46,47)(51,54);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(27,37)(28,40)(31,46)(32,48)(35,51)(38,44)(41,52)(49,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,33)(31,46)(32,48)(34,42)(35,51)(45,47)(49,56)(50,54)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di

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, 6405414);