#9236828

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

5288 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9144024);