#8944523

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

5025 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8851719);