#9075978

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

4770 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8983174);