#6437601

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

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

5930 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)(7,19)(8,23)(9,15)(10,16)(11,17)(12,18)(13,21)(14,27)(20,30)(22,29)(24,31)(25,36)(26,33)(28,34)(32,35);(2,4)(5,7)(6,13)(8,24)(9,18)(10,15)(11,21)(12,22)(14,31)(16,19)(17,29)(20,26)(23,30)(25,34)(27,36)(28,32)(33,35);(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,28)(13,30)(15,32)(16,23)(17,34)(18,33)(19,35)(21,24)(22,36)(29,31);(5,9)(6,15)(7,17)(8,25)(10,29)(11,19)(12,13)(14,32)(16,22)(18,21)(20,34)(23,36)(24,33)(26,31)(27,35)(28,30);(5,10,12,11)(6,16,18,17)(7,15,22,21)(8,26,28,27)(9,29,13,19)(14,23,33,34)(20,32,36,24)(25,31,30,35)

Polycyclic presentation:

2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6344797);