#123851

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1439
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)(7,17,19,28,20,12,26,29)(8,21,23,13,25,32,24,31)(18,30,34,33,35,36,22,27);(5,7,10,19,11,20,9,26)(6,12,15,29,14,17,16,28)(8,22,24,18,25,34,23,35)(13,30,21,33,31,36,32,27);(2,4)(5,8)(6,13,14,31)(7,18)(9,23)(10,24)(11,25)(12,27,17,33)(15,21,16,32)(19,34)(20,35)(22,26)(28,36,29,30)

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,f^a=fi,c^b=cf,e^b=ei,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, 31047);