#6573649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,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, 6480845);