#10147830

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

4749 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)(3,5)(4,6)(7,8)(10,14)(12,16)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,31,32,29);(1,3,7,6)(2,4,8,5)(10,14)(11,15)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31);(1,4,7,5)(2,3,8,6)(9,10,13,14)(11,12,15,16);(2,8)(4,5)(10,14)(11,15)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32);(1,5,7,4)(2,3,8,6)(9,11)(10,12)(13,15)(14,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28);(1,6,7,3)(2,5,8,4)(9,12)(10,15)(11,14)(13,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10055026);