#10349127

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

5129 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10256323);