#6561707

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=g,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei,e^d=eh

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, 6468903);