#9062219

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

4933 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)(11,13)(12,16)(14,15)(17,18,22,23)(19,26,27,24)(20,25,30,21)(28,32,31,29);(1,2)(3,7)(4,5)(6,8)(10,15)(11,16)(18,23)(19,27)(20,30)(29,32);(2,5)(3,8)(9,11,14,16)(10,12,15,13)(19,27)(20,30)(21,25)(24,26);(1,3,4,8)(2,6,5,7)(9,12,14,13)(10,11,15,16)(17,19)(18,24)(20,31)(21,29)(22,27)(23,26)(25,32)(28,30);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,16)(14,15)(17,20)(18,25)(19,28)(21,23)(22,30)(24,29)(26,32)(27,31);(1,2)(3,6)(4,5)(7,8)(9,13,14,12)(10,16,15,11)(17,21,22,25)(18,20,23,30)(19,29,27,32)(24,31,26,28)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969415);