#6630080

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

2994 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)(7,17)(8,22)(9,16)(10,13)(11,15)(12,20)(14,23)(18,32)(19,28)(21,26)(24,30)(25,29)(27,33)(31,34)(35,36);(5,7,9,20,10,19,11,21)(6,12,15,17,13,26,16,28)(8,18,23,34,24,33,25,35)(14,27,22,31,29,32,30,36);(1,2,3,4)(6,13)(12,26)(14,29)(15,16)(17,28)(22,30)(27,32)(31,36);(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,27)(13,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(26,32)(28,36);(7,19)(12,26)(17,28)(18,33)(20,21)(27,32)(31,36)(34,35)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,e^b=ei,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, 6537276);