#6583382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2199 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)(15,20)(16,29)(17,22)(18,23)(19,24)(21,30)(25,43)(26,36)(27,37)(28,38)(31,40)(32,35)(33,41)(34,42)(39,44);(6,9)(10,12)(13,15,18,26)(14,20,23,36)(16,27,30,44)(17,32,31,25)(19,34,33,28)(21,39,29,37)(22,35,40,43)(24,42,41,38);(1,2,3,4)(13,16)(14,21)(15,25)(17,19)(18,30)(20,35)(22,41)(23,29)(24,40)(26,32)(27,34)(28,44)(31,33)(36,43)(37,38)(39,42);(5,7,8,11)(6,10,9,12)(15,26)(16,17)(19,33)(20,36)(21,40)(22,29)(24,41)(25,27)(30,31)(32,44)(35,39)(37,43);(13,17,18,31)(14,22,23,40)(15,27,26,44)(16,19,30,33)(20,37,36,39)(21,41,29,24)(25,34,32,28)(35,38,43,42)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490578);