#6629741

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
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

2308 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)(17,23)(18,21)(19,24)(20,27)(22,34)(26,36)(28,37)(29,35)(31,39)(32,40)(33,38)(41,43)(42,44);(5,7,8,11)(6,9,10,12)(13,15,19,29,18,28,17,27)(14,20,23,37,21,35,24,25)(16,31,34,42,32,26,33,43)(22,39,30,41,38,36,40,44);(6,10)(9,12)(14,21)(16,32)(20,35)(23,24)(25,37)(26,31)(33,34)(42,43);(2,4)(13,16,17,33,18,32,19,34)(14,22,24,40,21,38,23,30)(15,26,27,42,28,31,29,43)(20,36,25,41,35,39,37,44);(13,17,18,19)(14,23,21,24)(15,27,28,29)(16,33,32,34)(20,37,35,25)(22,30,38,40)(26,42,31,43)(36,44,39,41)

Polycyclic presentation:

2^9|b^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,d^b=di,d^c=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, 6536937);