#8630868

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

4810 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)(11,12)(13,16)(14,15)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29);(1,3,6,5)(2,4,7,8)(9,11)(10,12)(13,14)(15,16)(18,23)(19,26)(25,28)(27,32);(2,7)(4,8)(9,12)(10,13)(11,15)(14,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(2,7)(3,5)(9,11,14,13)(10,12,15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,12,15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,5,6,3)(2,4,7,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=gh,e^2=i,f^2=g,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, 8538064);