#9779733

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

4597 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)(11,16)(12,15)(13,14)(17,18,22,23)(19,24,26,20)(21,29,31,25)(27,28,32,30);(2,5)(6,7)(10,14)(15,16)(17,18,22,23)(19,24,26,20)(21,25,31,29)(27,30,32,28);(1,2)(3,6)(4,5)(7,8)(18,23)(19,26)(25,29)(27,32);(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,30)(28,29)(31,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,32,29,27);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,29)(24,30)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=h,f^2=g,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, 9686929);