#8384234

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

5069 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8291430);