#8139649

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

4872 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,31)(21,29)(22,28)(23,26)(24,30)(25,32);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,18,23,26)(19,24,30,27)(20,31,28,22)(21,25,32,29);(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,26,27)(20,21,28,32)(22,29,31,25);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(17,20)(18,22)(19,21)(23,28)(24,29)(25,27)(26,31)(30,32);(1,2)(3,5)(4,6)(7,8)(17,21)(18,25)(19,28)(20,30)(22,24)(23,32)(26,29)(27,31);(1,3)(2,5)(4,8)(6,7)(17,22)(18,20)(19,29)(21,24)(23,31)(25,30)(26,28)(27,32)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046845);