#8990732

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

4957 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=gi,e^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, 8897928);