#8230864

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

4926 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8138060);