#6629477

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
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	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2360 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)(7,10)(8,9)(11,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,33)(26,35)(27,37)(29,36)(31,38)(32,40)(34,39)(41,43)(42,44);(6,9)(10,12)(13,15,18,29,19,27,17,28)(14,20,22,37,24,36,23,25)(16,31,33,42,34,26,32,43)(21,38,30,41,40,35,39,44);(5,7,8,11)(6,10,9,12)(13,16,17,32,19,34,18,33)(14,21,23,39,24,40,22,30)(15,26,28,42,27,31,29,43)(20,35,25,41,36,38,37,44);(2,4)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43);(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,32,34,33)(20,37,36,25)(21,30,40,39)(26,42,31,43)(35,44,38,41)

Polycyclic presentation:

2^9|b^2=g,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=ci,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536673);