#8969387

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5335 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8876583);