#6437220

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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,26,30)(23,31,35,45)(24,28,36,42)(25,38,32,29)(27,39,41,34)(33,44,47,52)(37,49,46,43)(40,50,51,48);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,32,39,26,35,25,34)(22,27,38,45,30,41,29,31)(24,33,46,50,36,47,37,48)(28,40,49,52,42,51,43,44);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,40)(29,43)(30,42)(31,44)(32,46)(34,48)(35,47)(38,49)(39,50)(41,51)(45,52);(6,11)(10,16)(12,18)(17,20)(24,36)(28,42)(33,47)(37,46)(40,51)(43,49)(44,52)(48,50);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^2=h,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, 6344416);