#10055646

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

5129 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)(11,15)(12,16)(17,18,22,25)(19,23,27,21)(20,24,28,31)(26,29,32,30);(1,2,4,6)(3,5,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,22,27)(18,21,25,23)(20,26,28,32)(24,30,31,29);(1,2,4,6)(3,5,8,7)(9,11,14,15)(10,12,13,16)(20,28)(24,31)(26,32)(29,30);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,32)(30,31);(1,2)(3,7)(4,6)(5,8)(9,13,14,10)(11,12,15,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,31)(27,32);(1,3,4,8)(2,5,6,7)(9,12)(10,15)(11,13)(14,16)(17,21,22,23)(18,19,25,27)(20,29,28,30)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=i,f^2=g,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, 9962842);