#123835

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1443
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1474 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=i,f^2=i,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,f^c=fi

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, 31031);