#391610

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=eg,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=dei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298806);