#8946361

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

4894 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8853557);