#8826310

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

5413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,e^2=gh,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=dgi,e^d=egi

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, 8733506);