#500718

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
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
prime radical	#61084	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1069 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)(7,17)(8,14)(9,26)(10,28)(11,16)(12,20)(13,22)(15,24)(18,32)(19,35)(21,36)(23,34)(25,33)(27,31)(29,30);(1,2,4,3)(7,18)(8,22)(9,25)(10,24)(12,30)(13,14)(15,28)(17,32)(19,23)(20,29)(21,27)(26,33)(31,36)(34,35);(5,7,11,19)(6,12,16,31)(8,23,25,18)(9,27,22,20)(10,29,24,21)(13,30,26,36)(14,17,33,35)(15,34,28,32);(1,3,4,2)(7,19)(12,31)(17,35)(18,23)(20,27)(21,29)(30,36)(32,34)

Polycyclic presentation:

2^9|b^2=e,c^2=i,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,d^c=di,f^c=fh,g^c=ghi,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, 407914);