#6687613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,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, 6594809);