#500713

order	512
nilpotency class	4
isoclinism	isoclinic class of #499757
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1045 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407909);