#6572211

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	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

1313 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)(3,7)(4,8)(5,6)(25,26,30,33)(27,31,36,45)(28,38,40,48)(29,41,43,34)(32,42,47,39)(35,49,50,56)(37,53,52,46)(44,54,55,51);(2,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,27,30,36)(26,31,33,45)(28,35,40,50)(29,37,43,52)(32,44,47,55)(34,46,41,53)(38,49,48,56)(39,51,42,54);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28,30,40)(26,32,33,47)(27,35,36,50)(29,42,43,39)(31,44,45,55)(34,48,41,38)(37,54,52,51)(46,56,53,49);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(27,36)(31,45)(35,50)(37,52)(44,55)(46,53)(49,56)(51,54);(1,4,5,3)(2,8,6,7)(12,20)(15,22)(18,24)(21,23)(26,33)(31,45)(32,47)(34,41)(38,48)(44,55)(46,53)(49,56)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,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, 6479407);