#8287769

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

5176 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)(17,18)(19,23)(20,27)(21,26)(22,32)(24,29)(25,31)(28,30);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,19)(18,25)(20,30)(21,22)(23,29)(24,31)(26,28)(27,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,26,29,27)(19,30,31,22)(23,32,25,28);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,21,24,20)(18,27,29,26)(19,30,31,22)(23,32,25,28);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,22,24,30)(18,28,29,32)(19,21,31,20)(23,27,25,26);(1,2)(3,6)(4,5)(7,8)(17,23,24,25)(18,19,29,31)(20,28,21,32)(22,27,30,26)

Polycyclic presentation:

2^9|c^2=g,d^2=gi,e^2=g,f^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, 8194965);