#500721

order	512
nilpotency class	4
isoclinism	isoclinic class of #499754
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	#61038	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1077 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)(7,16)(8,21)(9,27)(10,15)(11,18)(12,25)(13,31)(14,24)(17,20)(19,33)(22,26)(23,34)(28,32)(29,36)(30,35);(1,2,4,3)(5,6)(7,17)(8,22)(9,27)(10,15)(11,18)(12,31)(13,25)(14,24)(16,20)(19,32)(21,26)(23,34)(28,33)(29,36)(30,35);(5,7,10,20)(6,11,15,30)(8,23,26,36)(9,28,24,19)(12,32,13,33)(14,34,27,29)(16,22,17,21)(18,31,35,25);(1,3,4,2)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,f^2=hi,g^2=hi,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=gh,h^b=hi,f^c=fh,g^c=gh,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, 407917);