#8301487

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

4829 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8208683);