#6630081

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

2970 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)(8,18)(9,21)(12,22)(14,24)(15,29)(17,32)(19,20)(23,27)(26,34)(30,31)(33,36);(5,6,9,11,10,14,21,24)(7,12,17,22,16,26,32,34)(8,13,19,23,20,28,18,27)(15,25,31,33,30,35,29,36);(1,2,3,4)(5,7)(6,12)(8,15)(9,17)(10,16)(11,22)(13,25)(14,26)(18,29)(19,31)(20,30)(21,32)(23,33)(24,34)(27,36)(28,35);(5,8)(6,13)(7,15)(9,19)(10,20)(11,23)(12,25)(14,28)(16,30)(17,31)(18,21)(22,33)(24,27)(26,35)(29,32)(34,36);(7,16)(12,26)(15,30)(17,32)(22,34)(25,35)(29,31)(33,36)

Polycyclic presentation:

2^9|b^2=f,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, 6537277);