#6370641

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=fg,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277837);