#6555163

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,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,e^b=ei,d^c=di,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, 6462359);