#9787113

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
prime radical	#43111	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4685 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9694309);