#506411

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
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	#33580	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=eg,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=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413607);