#6437801

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

4508 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)(5,6,12,18,11,7,10,15)(8,16,23,19,21,13,22,29)(9,14,27,30,26,17,25,24)(20,31,35,32,33,28,34,36);(5,7,11,6)(8,19,21,29)(9,24,26,30)(10,15,12,18)(13,23,16,22)(14,27,17,25)(20,31,33,28)(32,35,36,34);(2,4)(5,8)(6,13)(7,16)(9,20)(10,22)(11,21)(12,23)(14,28)(15,19)(17,31)(18,29)(24,32)(25,34)(26,33)(27,35)(30,36);(5,9)(6,14)(7,17)(8,20)(10,25)(11,26)(12,27)(13,28)(15,24)(16,31)(18,30)(19,32)(21,33)(22,34)(23,35)(29,36);(8,21)(13,16)(19,29)(20,33)(22,23)(28,31)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=fi,b^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, 6344997);