#6583198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574075
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

2974 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)(5,6,11,17)(7,18,8,13)(9,14,27,33)(10,16,22,15)(12,21,29,20)(19,35,23,30)(24,28,36,31)(25,34,26,32);(5,7,10,20,11,8,22,21)(6,12,15,18,17,29,16,13)(9,23,25,36,27,19,26,24)(14,31,32,30,33,28,34,35);(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)(27,36)(30,34)(31,33)(32,35);(5,9)(6,14)(7,19)(8,23)(10,25)(11,27)(12,28)(13,30)(15,32)(16,34)(17,33)(18,35)(20,24)(21,36)(22,26)(29,31);(5,8)(6,13)(7,11)(9,24)(10,21)(12,15)(14,28)(16,29)(17,18)(19,26)(20,22)(23,25)(27,36)(30,34)(31,33)(32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490394);