#10058483

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

4926 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=g,e^2=hi,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, 9965679);