#6583202

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,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, 6490398);