#6687601

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1722 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)(25,28)(26,30)(27,34)(29,37)(32,40)(33,41)(35,44)(38,49)(39,42)(43,50)(45,48)(46,52)(47,51);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,33)(25,34)(26,32)(28,31)(29,41)(30,44)(35,39)(36,45)(37,48)(38,46)(40,42)(43,52)(47,50)(49,51);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(22,28)(23,32)(24,37)(26,39)(27,40)(31,44)(33,47)(34,35)(41,52)(43,49)(45,51)(46,48);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24,25,37)(22,29,28,36)(23,33,34,48)(26,38,39,50)(27,41,31,45)(30,43,42,49)(32,46,35,47)(40,51,44,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(22,28)(23,32)(26,39)(27,40)(29,36)(31,44)(33,46)(34,35)(38,50)(41,51)(45,52)(47,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,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, 6594797);