#6552136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459332);