#6454777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3524	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361973);