#6671497

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2161 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)(6,9)(7,11)(13,14)(15,24)(16,29)(17,22)(18,23)(19,21)(20,28)(25,36)(26,34)(27,41)(30,40)(31,44)(32,39)(33,37)(35,43)(38,42);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,26)(17,31)(18,28)(19,25)(21,34)(22,38)(23,24)(27,30)(29,36)(32,42)(33,43)(35,40)(37,41)(39,44);(2,4)(15,25)(16,19)(20,34)(21,29)(24,36)(26,28)(27,31)(30,33)(35,38)(37,40)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,21,23,29)(15,26,28,25)(17,30,32,33)(20,34,24,36)(22,37,39,40)(27,42,43,31)(35,38,41,44);(6,9)(10,12)(13,17,18,32)(14,22,23,39)(15,27,28,43)(16,30,19,33)(20,35,24,41)(21,37,29,40)(25,31,26,42)(34,38,36,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578693);