#391689

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

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

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=fg,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,d^c=de

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, 298885);