#6672337

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,26)(23,28,33,39)(24,29,36,40)(25,30,38,41)(31,42,46,34)(32,43,47,37)(35,44,49,50)(45,51,52,48);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,23,27,33)(22,28,26,39)(24,34,36,42)(25,37,38,43)(29,31,40,46)(30,32,41,47)(35,45,49,52)(44,51,50,48);(2,4)(21,24,27,36)(22,29,26,40)(23,31,33,46)(25,35,38,49)(28,42,39,34)(30,44,41,50)(32,45,47,52)(37,48,43,51);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(25,38)(30,41)(32,47)(35,49)(37,43)(44,50)(45,52)(48,51);(7,15)(10,13)(14,20)(18,19)(21,25,27,38)(22,30,26,41)(23,32,33,47)(24,35,36,49)(28,43,39,37)(29,44,40,50)(31,45,46,52)(34,48,42,51)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^2=i,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, 6579533);