#8531710

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

4961 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^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, 8438906);