#10261478

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

5106 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)(3,5,4,7)(17,18)(19,26)(20,29)(21,24)(22,23)(25,30)(27,31)(28,32);(1,3,6,4)(2,7,8,5)(9,10)(11,12)(13,15)(14,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32);(9,11)(10,14)(12,15)(13,16)(17,20,23,30)(18,25,22,29)(19,27,21,28)(24,31,26,32);(2,8)(3,4)(9,10,13,15)(11,14,16,12)(17,20)(18,25)(19,28)(21,27)(22,29)(23,30)(24,32)(26,31);(1,4,6,3)(2,7,8,5)(9,10)(11,14)(12,16)(13,15)(17,21,23,19)(18,24,22,26)(20,28,30,27)(25,31,29,32);(1,5,6,7)(2,4,8,3)(9,12,13,14)(10,11,15,16)(17,22,23,18)(19,26,21,24)(20,29,30,25)(27,32,28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10168674);