#9836826

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

4587 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=hi,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, 9744022);