#8509559

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

5030 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416755);