#8969531

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

4751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=gh,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, 8876727);