#6433259

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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×#3050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,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, 6340455);