#6442955

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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

2586 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,20)(18,21)(19,31)(22,23)(24,30)(25,28)(26,29)(27,33)(32,35)(34,36);(2,4)(5,6)(7,18)(8,22)(9,15)(10,14)(11,16)(12,20)(13,23)(17,21)(19,27)(24,29)(25,28)(26,30)(31,33)(32,34)(35,36);(5,7,10,25,11,21,9,20)(6,12,14,18,16,28,15,17)(8,19,26,34,24,35,23,33)(13,27,22,32,30,36,29,31);(5,8)(6,13)(7,19)(9,23)(10,26)(11,24)(12,27)(14,22)(15,29)(16,30)(17,31)(18,32)(20,33)(21,35)(25,34)(28,36);(5,9,11,10)(6,14,16,15)(7,20,21,25)(8,23,24,26)(12,18,28,17)(13,22,30,29)(19,33,35,34)(27,32,36,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350151);