#123787

order	512
nilpotency class	3
isoclinism	isoclinic class of #2104
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#401
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1443
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30983);