#6630077

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2982 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)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,23)(18,28)(19,33)(21,26)(24,29)(25,30)(27,32)(31,34)(35,36);(5,7,9,20,11,18,10,21)(6,12,15,17,16,26,14,28)(8,19,23,34,24,32,25,35)(13,27,22,31,30,33,29,36);(1,2,3,4)(7,18)(12,26)(17,28)(19,32)(20,21)(27,33)(31,36)(34,35);(5,8)(6,13)(7,19)(9,23)(10,25)(11,24)(12,27)(14,29)(15,22)(16,30)(17,31)(18,32)(20,34)(21,35)(26,33)(28,36);(5,9,11,10)(6,14,16,15)(7,20,18,21)(8,23,24,25)(12,28,26,17)(13,29,30,22)(19,34,32,35)(27,36,33,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537273);