#6693957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1194 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)(3,6,4,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,34)(27,31,38,47)(28,39,41,32)(29,33,43,50)(35,44,37,46)(36,48,40,45)(42,53,54,49)(51,56,52,55);(1,3,5,4)(2,6,7,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,31)(28,40)(29,35)(30,38)(32,48)(33,44)(34,47)(36,41)(37,43)(39,45)(42,52)(46,50)(49,56)(51,54)(53,55);(1,4,5,3)(2,6,7,8)(27,35)(28,41)(29,43)(31,44)(32,39)(33,50)(36,52)(37,38)(40,51)(45,56)(46,47)(48,55);(10,15)(14,17)(16,23)(22,24)(25,28,30,41)(26,32,34,39)(27,36,38,40)(29,42,43,54)(31,45,47,48)(33,49,50,53)(35,51,37,52)(44,55,46,56);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,35)(29,43)(31,44)(33,50)(36,51)(37,38)(40,52)(42,54)(45,55)(46,47)(48,56)(49,53)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6601153);