#6674278

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 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)(21,22,27,32)(23,28,36,44)(24,29,39,47)(25,30,26,31)(33,41,50,52)(34,42,35,43)(37,45,49,51)(38,46,40,48);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23,27,36)(22,28,32,44)(24,37,39,49)(25,34,26,35)(29,45,47,51)(30,42,31,43)(33,40,50,38)(41,48,52,46);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24)(22,29)(23,33)(25,38)(26,40)(27,39)(28,41)(30,46)(31,48)(32,47)(34,49)(35,37)(36,50)(42,51)(43,45)(44,52);(6,13)(11,15)(12,14)(19,20)(21,25,27,26)(22,30,32,31)(23,34,36,35)(24,38,39,40)(28,42,44,43)(29,46,47,48)(33,49,50,37)(41,51,52,45);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(23,35)(25,26)(28,43)(30,31)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,50)(51,52)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6581474);