#10040523

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

4689 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,18,23,29)(19,24,28,22)(20,27,31,25)(21,32,30,26);(1,2,4,6)(3,5,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,28)(18,24,29,22)(20,21,31,30)(25,26,27,32);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,16)(13,15)(17,20,23,31)(18,25,29,27)(19,30,28,21)(22,26,24,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21,23,30)(18,26,29,32)(19,20,28,31)(22,27,24,25);(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,15)(14,16)(17,20)(18,27)(19,30)(21,28)(22,32)(23,31)(24,26)(25,29);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,16)(13,15)(17,22,23,24)(18,28,29,19)(20,26,31,32)(21,27,30,25)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,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, 9947719);