#6694118

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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)(27,31,38,46)(28,39,40,32)(29,33,42,48)(35,43,37,45)(36,50,52,44)(41,53,54,47)(49,56,51,55);(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)(28,36)(29,35)(30,38)(32,44)(33,43)(34,46)(37,42)(39,50)(40,52)(41,49)(45,48)(47,55)(51,54)(53,56);(1,4,5,3)(2,6,7,8)(27,35)(28,40)(29,42)(31,43)(32,39)(33,48)(36,51)(37,38)(44,56)(45,46)(49,52)(50,55);(10,15)(14,17)(16,23)(22,24)(25,28,30,40)(26,32,34,39)(27,36,38,52)(29,41,42,54)(31,44,46,50)(33,47,48,53)(35,49,37,51)(43,55,45,56);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,35)(29,42)(31,43)(33,48)(36,49)(37,38)(41,54)(44,55)(45,46)(47,53)(50,56)(51,52)

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,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, 6601314);