#10047063

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

5125 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=ghi,e^2=h,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, 9954259);