#8760375

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

5059 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)(19,24)(20,26)(21,27)(22,32)(23,25)(28,31)(29,30);(1,2)(3,7)(4,5)(6,8)(10,14)(12,15)(17,19,23,20)(18,24,25,26)(21,30,32,28)(22,31,27,29);(2,5)(6,7)(9,10,13,14)(11,15,16,12)(18,25)(19,20)(21,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,23,19)(18,26,25,24)(21,30,32,28)(22,31,27,29);(1,2)(3,6)(4,5)(7,8)(9,12,13,15)(10,11,14,16)(17,21)(18,27)(19,28)(20,30)(22,25)(23,32)(24,31)(26,29);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,22,23,27)(18,21,25,32)(19,29,20,31)(24,28,26,30)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=gh,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, 8667571);