#6576191

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1948 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)(5,6,8,9)(7,10,11,12)(13,14,19,24)(15,20,27,36)(16,21,17,22)(18,23,33,42)(25,34,43,44)(26,35,32,41)(28,37,30,39)(29,38,31,40);(6,9)(10,12)(13,15)(14,20)(16,28)(17,30)(18,32)(19,27)(21,37)(22,39)(23,41)(24,36)(25,29)(26,33)(31,43)(34,38)(35,42)(40,44);(2,4)(13,16,19,17)(14,21,24,22)(15,25,27,43)(18,31,33,29)(20,34,36,44)(23,40,42,38)(26,30,32,28)(35,39,41,37);(5,7,8,11)(6,10,9,12)(16,29)(17,31)(18,33)(21,38)(22,40)(23,42)(25,28)(26,32)(30,43)(34,37)(35,41)(39,44);(13,17,19,16)(14,22,24,21)(15,25,27,43)(18,29,33,31)(20,34,36,44)(23,38,42,40)(26,30,32,28)(35,39,41,37)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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, 6483387);