#6573651

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

4409 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)(7,18)(8,14)(9,15)(10,19)(11,16)(12,17)(13,21)(20,35)(22,24)(23,30)(25,31)(26,33)(27,34)(28,32)(29,36);(5,7,12,22)(6,13,17,8)(9,20,27,31)(10,14,11,21)(15,29,34,23)(16,24,19,18)(25,33,35,32)(26,30,28,36);(2,4)(5,8,12,13)(6,14,17,21)(7,19,22,16)(9,25,27,35)(10,24,11,18)(15,31,34,20)(23,26,29,28)(30,33,36,32);(5,9,12,27)(6,15,17,34)(7,20,22,31)(8,23,13,29)(10,26,11,28)(14,30,21,36)(16,32,19,33)(18,35,24,25);(5,8,12,13)(6,14,17,21)(7,19,22,16)(9,25,27,35)(10,24,11,18)(15,31,34,20)(23,26,29,28)(30,33,36,32)

Polycyclic presentation:

2^9|b^2=i,c^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=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, 6480847);