#6674113

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	#42908	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1835 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)(23,33)(24,29)(25,38)(26,31)(27,32)(28,36)(30,40)(34,42)(35,43)(37,46)(39,45)(41,44)(47,48)(49,52)(50,51);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23)(22,28)(24,35)(25,34)(26,36)(27,41)(29,44)(30,42)(31,33)(32,43)(37,49)(38,47)(39,50)(40,48)(45,52)(46,51);(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)(30,46)(31,32)(34,48)(35,41)(37,40)(38,45)(42,47)(43,44);(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)(29,45,32,46)(35,49,41,50)(43,51,44,52);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(22,31)(23,35)(24,27)(28,43)(30,38)(33,44)(34,49)(36,41)(37,39)(42,51)(47,52)(48,50)

Polycyclic presentation:

2^9|c^2=fi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6581309);