#403916

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2768 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)(3,4)(5,6)(7,8)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(2,4)(10,13)(14,16)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,29,28,32);(6,8)(9,11,12,15)(10,14,13,16)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311112);