#6677312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22)(23,27)(24,28)(25,30)(26,29)(31,40)(32,47)(33,41)(34,42)(35,43)(36,44)(37,45)(38,48)(39,46)(49,51)(50,52);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,26,34)(22,27,29,42)(24,35,36,50)(25,32,39,33)(28,43,44,52)(30,47,46,41)(31,38,49,37)(40,48,51,45);(2,4)(21,24,26,36)(22,28,29,44)(23,31,34,49)(25,38,39,37)(27,40,42,51)(30,48,46,45)(32,35,33,50)(41,52,47,43);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(24,36)(28,44)(31,49)(35,50)(37,38)(40,51)(43,52)(45,48);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,29)(23,32)(25,39)(27,41)(28,44)(31,35)(33,34)(37,38)(40,52)(42,47)(43,51)(49,50)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=f,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,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584508);