#10040627

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

4515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947823);