#9065829

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

5023 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,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, 8973025);