#501766

order	512
nilpotency class	4
isoclinism	isoclinic class of #501766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

547 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:

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

Polycyclic presentation:

2^9|a^2=efh,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,f^a=fi,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, 408962);