#6583626

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 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)(5,6,8,9)(7,10,11,12)(13,14,19,24)(15,20,26,35)(16,21,32,41)(17,22,30,39)(18,23,31,40)(25,34,43,44)(27,36,29,38)(28,37,33,42);(6,9)(10,12)(13,15)(14,20)(16,29)(17,25)(18,33)(19,26)(21,38)(22,34)(23,42)(24,35)(27,32)(28,31)(30,43)(36,41)(37,40)(39,44);(1,2,3,4)(13,16)(14,21)(15,25)(17,31)(18,30)(19,32)(20,34)(22,40)(23,39)(24,41)(26,43)(27,28)(29,33)(35,44)(36,37)(38,42);(5,7,8,11)(6,10,9,12)(15,26)(16,30)(17,32)(18,31)(20,35)(21,39)(22,41)(23,40)(25,29)(27,43)(34,38)(36,44);(13,17,19,30)(14,22,24,39)(15,27,26,29)(16,31,32,18)(20,36,35,38)(21,40,41,23)(25,28,43,33)(34,37,44,42)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,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, 6490822);