#6559082

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 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)(7,15)(8,17)(9,13)(10,12)(11,14)(16,19)(18,20)(21,22)(23,32)(24,29)(25,30)(26,40)(27,31)(28,35)(33,45)(34,42)(36,46)(37,43)(38,44)(39,50)(41,49)(47,52)(48,51);(5,7,11,10)(6,12,14,15)(8,18,9,16)(13,20,17,19)(21,23,26,35,27,36,25,34)(22,28,30,32,31,42,40,46)(24,33,39,49,37,47,38,48)(29,41,44,45,43,51,50,52);(1,2,3,4)(5,8,11,9)(6,13,14,17)(7,16,10,18)(12,19,15,20)(21,24,27,37)(22,29,31,43)(23,33,36,47)(25,38,26,39)(28,41,42,51)(30,44,40,50)(32,45,46,52)(34,48,35,49);(5,9,11,8)(6,13,14,17)(7,16,10,18)(12,20,15,19)(24,37)(29,43)(33,47)(38,39)(41,51)(44,50)(45,52)(48,49);(5,10,11,7)(6,12,14,15)(8,18,9,16)(13,19,17,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,38,37,39)(28,32,42,46)(29,44,43,50)(33,48,47,49)(41,45,51,52)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6466278);