#8765603

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

5273 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8672799);