#8158650

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

4880 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8065846);