#6583194

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3014 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:

(2,4)(5,6)(7,18)(8,13)(9,14)(10,16)(11,17)(12,20)(15,22)(19,34)(21,29)(23,30)(24,28)(25,33)(26,31)(27,32)(35,36);(5,7,10,20,11,8,22,21)(6,12,15,18,17,29,16,13)(9,19,25,24,27,23,26,35)(14,28,31,34,32,36,33,30);(1,2,3,4)(5,8)(6,13)(7,11)(9,24)(10,21)(12,15)(14,28)(16,29)(17,18)(19,26)(20,22)(23,25)(27,35)(30,33)(31,34)(32,36);(5,9,11,27)(6,14,17,32)(7,19,8,23)(10,25,22,26)(12,28,29,36)(13,30,18,34)(15,31,16,33)(20,24,21,35);(5,8)(6,13)(7,11)(9,24)(10,21)(12,15)(14,28)(16,29)(17,18)(19,26)(20,22)(23,25)(27,35)(30,33)(31,34)(32,36)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6490390);