#6437792

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4615 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)(5,6,9,14,11,16,10,15)(7,17,20,28,21,27,19,12)(8,13,23,30,24,31,25,32)(18,29,22,35,34,36,33,26);(5,7)(6,12)(8,22)(9,19)(10,20)(11,21)(13,29)(14,27)(15,17)(16,28)(18,23)(24,33)(25,34)(26,30)(31,36)(32,35);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,32)(16,31)(17,29)(19,33)(20,22)(21,34)(27,36)(28,35);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,32)(16,31)(17,29)(19,33)(20,22)(21,34)(27,36)(28,35);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,24,25)(12,27,28,17)(13,30,31,32)(18,33,34,22)(26,36,35,29)

Polycyclic presentation:

2^9|a^2=f,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,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344988);