#9719218

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

5028 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)(3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(1,3)(2,6)(4,7)(5,8)(10,14)(11,15)(19,25)(23,29)(26,31)(30,32);(9,11,13,15)(10,12,14,16)(18,22)(23,29)(24,27)(30,32);(9,12)(10,15)(11,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(1,4,5,6)(2,3,7,8)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,3)(2,4)(5,8)(6,7)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)

Polycyclic presentation:

2^9|c^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9626414);