#6573642

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

4413 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)(7,20)(8,15)(9,16)(10,17)(11,22)(12,18)(13,19)(14,23)(21,35)(24,30)(25,31)(26,33)(27,34)(28,32)(29,36);(5,7,13,15)(6,14,19,10)(8,22,20,18)(9,21,27,30)(11,17,12,23)(16,29,34,25)(24,33,35,32)(26,31,28,36);(2,4)(5,8)(6,15)(7,19)(9,25)(10,12)(11,14)(13,20)(16,31)(17,18)(21,32)(22,23)(24,26)(27,29)(28,35)(30,33)(34,36);(5,9,13,27)(6,16,19,34)(7,21,15,30)(8,24,20,35)(10,25,14,29)(11,26,12,28)(17,31,23,36)(18,32,22,33);(5,10,13,14)(6,17,19,23)(7,22,15,18)(8,12,20,11)(9,24,27,35)(16,30,34,21)(25,26,29,28)(31,33,36,32)

Polycyclic presentation:

2^9|b^2=i,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=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480838);