#6686761

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1913 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593957);