#10042519

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

4695 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)(3,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,25)(19,24,20,30)(21,26,27,22)(28,29,31,32);(1,3)(2,5)(4,8)(6,7)(9,10,12,14)(11,13,16,15)(17,19)(18,24)(20,23)(21,31)(22,29)(25,30)(26,32)(27,28);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(18,25)(19,20)(21,27)(29,32);(1,2)(3,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,20,23,19)(18,24,25,30)(21,31,27,28)(22,32,26,29);(1,2)(3,5)(4,7)(6,8)(17,21)(18,26)(19,28)(20,31)(22,25)(23,27)(24,29)(30,32);(1,3)(2,6)(4,8)(5,7)(17,22,23,26)(18,27,25,21)(19,29,20,32)(24,28,30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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, 9949715);