#500726

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
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	#61089	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1092 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)(7,17)(8,22)(9,26)(10,25)(12,30)(13,14)(15,28)(18,34)(19,20)(21,27)(23,33)(24,29)(31,36)(32,35);(1,2,4,3)(5,6)(7,18)(8,23)(9,13)(10,28)(11,16)(12,19)(14,26)(15,25)(17,34)(20,30)(21,36)(22,33)(24,32)(27,31)(29,35);(5,7,11,21)(6,12,16,32)(8,24,26,19)(9,27,22,17)(10,29,25,20)(13,18,33,36)(14,30,23,35)(15,34,28,31);(1,3,4,2)

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=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,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, 407922);