#8599313

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

5047 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506509);