#8284184

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

5160 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)(3,7)(11,15)(13,16)(17,18,22,23)(19,25,26,21)(20,28,31,24)(27,30,29,32);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(24,28)(27,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(21,25)(24,28)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,29)(22,26)(23,25)(24,32)(27,31)(28,30);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,31)(18,24,23,28)(19,27,26,29)(21,30,25,32);(17,21)(18,19)(20,30)(22,25)(23,26)(24,27)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8191380);