#6437803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4476 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)(5,6,10,15,11,16,31,36)(7,17,21,33,22,32,20,12)(8,19,25,34,26,13,24,27)(9,14,28,23,29,18,30,35);(5,7,11,22)(6,12,16,33)(8,23,26,35)(9,27,29,34)(10,20,31,21)(13,30,19,28)(14,24,18,25)(15,32,36,17);(2,4)(5,8)(6,13)(7,18)(9,12)(10,25)(11,26)(14,22)(15,27)(16,19)(17,28)(20,23)(21,35)(24,31)(29,33)(30,32)(34,36);(5,9)(6,14)(7,19)(8,12)(10,28)(11,29)(13,22)(15,23)(16,18)(17,25)(20,27)(21,34)(24,32)(26,33)(30,31)(35,36);(5,10,11,31)(6,15,16,36)(7,20,22,21)(8,24,26,25)(9,28,29,30)(12,32,33,17)(13,34,19,27)(14,23,18,35)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=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, 6344999);