#6674725

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1790 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6581921);