#6436373

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(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)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(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)(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,37)(28,41)(30,44)(31,43)(32,45)(34,48)(35,49)(36,47)(40,50)(42,51)(46,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,37)(29,43)(33,47)(38,39)(41,51)(44,50)(45,52)(48,49);(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|a^2=h,b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343569);