#6673000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#23965	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 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)(15,19)(16,20)(17,22)(18,21)(23,32)(24,39)(25,33)(26,34)(27,35)(28,37)(29,36)(30,40)(31,38)(41,44)(42,43);(6,9)(7,11)(13,15,18,26)(14,19,21,34)(16,27,29,42)(17,24,31,25)(20,35,36,43)(22,39,38,33)(23,30,41,28)(32,40,44,37);(2,4)(13,16,18,29)(14,20,21,36)(15,23,26,41)(17,30,31,28)(19,32,34,44)(22,40,38,37)(24,27,25,42)(33,43,39,35);(6,9)(10,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,32)(24,39)(25,33)(26,34)(27,35)(28,37)(29,36)(30,40)(31,38)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(14,21)(15,24)(17,31)(19,33)(20,36)(23,27)(25,26)(28,30)(32,43)(34,39)(35,44)(41,42)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6580196);