#146624

order	512
nilpotency class	4
isoclinism	isoclinic class of #12767
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1790
commutator	C₂×D₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10745
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ehi,f^b=fh,d^c=dh,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53820);