#6494958

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402154);