#6437902

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4523 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)(5,6,10,15,12,17,11,16)(7,18,13,24,21,23,20,8)(9,14,26,30,28,32,27,31)(19,29,25,36,34,35,33,22);(2,4)(5,7)(6,8)(9,25)(10,20)(11,13)(12,21)(14,29)(15,23)(16,18)(17,24)(19,26)(22,30)(27,34)(28,33)(31,36)(32,35);(5,8)(6,13)(7,15)(9,22)(10,23)(11,18)(12,24)(14,25)(16,21)(17,20)(19,30)(26,35)(27,29)(28,36)(31,34)(32,33);(5,9)(6,14)(7,19)(8,22)(10,26)(11,27)(12,28)(13,25)(15,30)(16,31)(17,32)(18,29)(20,33)(21,34)(23,35)(24,36);(5,10,12,11)(6,15,17,16)(7,20,21,13)(8,23,24,18)(9,26,28,27)(14,30,32,31)(19,33,34,25)(22,35,36,29)

Polycyclic presentation:

2^9|a^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,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, 6345098);