#9788907

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	#45267	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,f^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, 9696103);