#6443337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4960 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:

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

Polycyclic presentation:

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

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, 6350533);