#6630082

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629034
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3018 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)(6,11)(8,19)(9,12)(13,22)(15,24)(16,30)(18,25)(20,21)(23,28)(27,34)(31,32)(33,36);(5,6,12,24,10,15,9,11)(7,13,25,34,17,27,18,22)(8,14,19,28,21,29,20,23)(16,26,30,36,31,35,32,33);(1,2,3,4)(5,7)(6,13)(8,16)(9,18)(10,17)(11,22)(12,25)(14,26)(15,27)(19,30)(20,32)(21,31)(23,33)(24,34)(28,36)(29,35);(5,8)(6,14)(7,16)(9,20)(10,21)(11,23)(12,19)(13,26)(15,29)(17,31)(18,32)(22,33)(24,28)(25,30)(27,35)(34,36);(7,17)(13,27)(16,31)(18,25)(22,34)(26,35)(30,32)(33,36)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,e^c=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, 6537278);