#8760575

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

4830 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,f^2=h,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, 8667771);