#6573639

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

4438 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:

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

Polycyclic presentation:

2^9|c^2=i,d^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, 6480835);