#407660

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

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, 314856);