#9210392

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

4928 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9117588);