#501652

order	512
nilpotency class	4
isoclinism	isoclinic class of #501652
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
prime radical	#33605	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

689 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408848);