#10086369

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

4931 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993565);