#8634841

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

4795 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)(3,8)(4,5)(6,7)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,23)(22,26)(27,30)(28,32)(29,31);(1,3,6,5)(2,4,7,8)(9,11,12,16)(10,13,14,15)(17,19,23,26)(18,24,21,22)(20,28,31,27)(25,32,29,30);(2,7)(4,8)(9,11)(10,13)(12,16)(14,15)(18,21)(19,26)(25,29)(27,28);(2,7)(3,5)(9,11)(10,13)(12,16)(14,15)(17,20,23,31)(18,25,21,29)(19,27,26,28)(22,32,24,30);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,21,23,18)(19,24,26,22)(20,29,31,25)(27,30,28,32);(1,5,6,3)(2,4,7,8)(17,22,23,24)(18,26,21,19)(20,30,31,32)(25,27,29,28)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8542037);