#6448855

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3198 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,e^c=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, 6356051);