#501656

order	512
nilpotency class	4
isoclinism	isoclinic class of #501655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408852);