#8946563

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

4752 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8853759);