#6694123

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gi,d^2=i,e^2=fi,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, 6601319);