#391959

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,21,23,24)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,23)(22,24)(25,27,29,35,28,34,36,40)(26,30,32,37,31,33,38,39);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(19,23)(21,24);(3,8)(5,6)(11,15)(13,16)(26,31)(30,33)(32,38)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=defi,d^c=dei

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, 299155);