#9434199

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

4699 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341395);