#6574076

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2528 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)(5,6,11,16)(7,13,8,21)(9,14,27,34)(10,17,22,15)(12,20,30,19)(18,31,23,36)(24,29,25,32)(26,35,28,33);(5,7,10,19,11,8,22,20)(6,12,15,13,16,30,17,21)(9,23,26,25,27,18,28,24)(14,32,33,36,34,29,35,31);(2,4)(5,8)(6,13)(7,11)(9,24)(10,20)(12,17)(14,32)(15,30)(16,21)(18,28)(19,22)(23,26)(25,27)(29,34)(31,35)(33,36);(5,9)(6,14)(7,18)(8,23)(10,26)(11,27)(12,29)(13,31)(15,33)(16,34)(17,35)(19,24)(20,25)(21,36)(22,28)(30,32);(5,7)(6,13)(8,11)(9,25)(10,19)(12,17)(14,32)(15,30)(16,21)(18,26)(20,22)(23,28)(24,27)(29,34)(31,35)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 6481272);